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Full text of "Experimental Nuclear Physics Vol 2"

mental 











Physics 



7c;LjiKw;; 



E. SEGRE 
Editor 



. 






EXPERIMENTAL 
NUCLEAR PHYSICS 



CONTRIBUTORS 

VOLUME I 
Julius Ashkin Kenneth T. Bainbridge 

Hans A. Bethe Norman F. Ramsey 

Hans H. Staub 

VOLUME II 
Philip Morrison Bernard T. Feld 

VOLUME III 
In preparation 




EXPERIMENTAL 
NUCLEAR PHYSICS 

VOLUME II 

E. SEGRE, Editor 
P. Morrison and B. T. Feld 



JOHN WILEY & SONS, INC., NEW YORK 
CHAPMAN & HALL, LIMITED, LONDON 



few. i £i>/ 



CLASSICS b,-^C NO. 

, ■„JjJ4^Ss _ 

Checked I ^te*V 



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/u-*>-70 



Copyright, 1953 

BY 

John Wiley & Sons, Inc. 



All Rights Reserved 

This book or any part thereof must not 
be reproduced in any form without 
the written permission of the publisher. 



Library of Congress Catalog Card Number: 52-5852 

PRINTED IN THE UNITED STATES OP AMERICA 



PREFACE 

At the end of World War II many physicists who had been mobilized 
for the war effort returned to university work and to pure research; a 
great number of them had worked on nuclear problems and were anxious 
to resume investigations in this field. Moreover there was a large influx 
of students eager to start nuclear investigations. 

The need was keenly felt for a book which would bring the experimen- 
talist up to date in experimental techniques, point out to him the sig- 
nificant facts and data, and indicate the broad lines of theoretical inter- 
pretation. 

It was immediately apparent that the field of nuclear physics had 
grown so much and the various branches had become so specialized that 
no one person could hope to write a book like the famous treatises of 
Rutherford (which, however, because of the evolution mentioned above, 
had by 1930 already become Rutherford, Chadwick, and Ellis), Curie, 
and Kohlrausch. A cooperative effort like the Geiger-Scheel Handbuch 
der Physik seemed the only solution. Individual authors could undertake 
to prepare reasonably complete treatises on a restricted field in which 
they are quite authoritative. By keeping the discussions relatively 
short, it became possible for a group of authors to cooperate without cur- 
tailing their research activity. An incentive for several of the authors, 
indeed, was the desire to read the contributions of the others. 

This work is the outcome of that effort. We hope that it will be useful 
to the serious student and to the research worker in the field. Each part, 
with its bibliography, should be sufficient to inform the reader about 
the main results obtained in nuclear physics up to the end of 1952 and 
enable him to go directly to the original literature or to the several 
excellent collections of periodical reviews which are currently appear- 
ing (Annual Reviews of Nuclear Science, Reviews of Modern Physics, 
Ergebnisse der exakten Naturwissenschaften, Progress in Nuclear Physics, 
and others) for further details. 

E. Segre 

Berkeley, California 

June, 1958 



CONTENTS 

Part VI. A Survey of Nuclear Reactions 

Philip Morrison 

1 . The Conservation Laws 1 
A. Application of Energy-Momentum Conservation. B. Conservation of 
Angular Momenta. 

2. The Data of Nuclear Reactions 15 
A. Yields and Cross Sections. B. The Measurement of Yields and Cross 
Sections. C. Types of Reactions: A Guide to the Literature. 

3. The Nuclear Model 25 
A. Qualitative Account of the Model. B. Calculation of Level Densities. 

4. Nuclear Level Widths 45 

A. Level Widths and Reaction Cross Sections: Statistical Relations. 

B. Calculation of Level Widths. 

5. The Course of Nuclear Reactions 54 
A. The Steps of the Reaction. B. The Contact Cross Section. C. The 
Disintegration of Compound States. 

6. The Dispersion Theory: Resonance Reactions 64 
A. The One-Body Model and Its Difficulties. B. The Dispersion Theory 

for an Isolated Resonance. C. The Generalized Theory of Dispersion: 
Many Levels and Many Decay Modes. D. Statistical Estimates. 

7. Some Typical Nuclear Reactions 83 
A. Resonance: The Region of Dispersion Theory. B. Reactions without 
Marked Resonance. 

8. The Deuteron as a Projectile 110 
A. The Oppenheimer-Phillips Reaction: Low-Energy Stripping. B. Strip- 
ping Reactions at Higher Energy. 

9. Radiative Processes in Nuclear Reactions 114 
A. The Multipole Classification. B. Calculation of Radiation Widths. 

C. Photo-Induced Reactions. 

10. Nuclear Fission 123 
A. The Energetics of Fission. B. The Products of Fission. C. Fission 
Cross Sections. 

11. Nuclear Reactions at High Energy 141 
A. The Nuclear Cascade. B. Correlations among Nucleons. C. The Op- 
tical Model for the Scattering of Nucleons. D. The Processes of Nuclear 
De-excitation at High Energy. E. Mesons: Virtual and Real. 

vii 



viii Contents 

Part VII. The Neutron 
Bernard T. Feld 

1. Properties and Fundamental Interactions 209 
A. Discovery. B. Properties. C. Fundamental Interactions. 

2. Interaction with Nuclei 247 
A. Introduction. B. General Considerations. C. Types of Neutron Re- 
actions. 

3. Sources and Detectors: Neutron Spectroscopy 357 
A. Introduction. B. Neutron Sources. C. Neutron Detectors. D. Slow- 
Neutron Spectroscopy. E. The Calibration of Neutron Sources. 

4. The Interaction of Neutrons with Matter in Bulk 460 
A. Introduction. B. Diffusion of Monoenergetic Neutrons. C. The 
Slowing-Down Process. D. The Nuclear Chain Reaction. 

5. Coherent Scattering Phenomena with Slow Neutrons 508 
A. Introduction. B. Neutron Diffraction and the Structure of Matter. 

C. Neutron Diffraction and the Determination of Nuclear Scattering 
Amplitudes. D. Magnetic Scattering and Neutron Polarization. 

Author Index 587 

Subject Index 595 



PART VI 



A Survey of Nuclear Reactions 

P. MORRISON 

Cornell University 

The study of nuclear reactions parallels the familiar study of chemical 
reactions. Most of the root ideas of chemistry are fundamental for the 
nuclear physicist. The equation of the reaction, the heat of reaction, 
the rate of reaction, the "balancing" of the equation — all these have 
their nuclear counterparts. The fact that the characteristic energy 
release is the millions of electron volts associated with the binding of 
nuclear particles, rather than the few volts which result from molecular 
binding, is the most striking inherent difference between the two fields. 
The nuclear physicist can detect the reaction not of moles or micromoles 
of his reactants, but of individual 'particles. This has meant that from 
the beginning the nuclear physicist has studied not equilibria and the 
laws of mass action, but the properties of single collisions. Just as the 
chemist has recently come to regard the individual collision as the center 
of attention for really fundamental understanding of reactions, so the 
physicist has now come to study large-scale reactions and even thermal 
equilibria in the nuclear domain in the light of his newer interest in 
astrophysical and chain reaction problems. Although the two points 
of view have tended to merge, it is our task to survey nuclear reactions 
primarily from the point of view of understanding individual collisions. 
Clearly this is the key to any genuine insight. 

SECTION 1. THE CONSERVATION LAWS 

Nuclear reactions, like chemical ones, can be more or less complex. 
The typical reaction — with which we shall be most concerned — involves 
a stationary or "target" nucleus bombarded by a relatively light incident 
nuclear projectile. The products of the reaction may be one, two, or 
even more nuclei. In the nuclear reactions of most interest before the 
development of the very high-energy accelerators in 1947, one to three 
products were by far most common. The target nucleus changes into a 
nuclear species differing by only a few units in A, the mass number or 

1 



2 A Survey of Nuclear Reactions [Pt. VI 

number of nucleons (protons and neutrons) in the nucleus, and in Z, the 
number of nuclear protons and, hence, the number of electrons in the 
neutral atom. Such reactions are designated either in the somewhat 
redundant notation taken from the chemists, an example of which might 
be uNa 23 + 1H 1 — * i2Mg 23 + on 1 [on 1 = neutron], or in an abbreviated 
scheme, due to Bothe. In the latter, one writes first the target nucleus, 
indicating its mass number and chemical character (often giving the 
nuclear charge Z as well), then in parentheses the symbols for the bom- 
barding and emitted particles in order, and finally the residual nucleus. 
The above reaction would be written Na 23 (p,n)Mg 23 . If there are 
more than two products of reaction, one might write, for example, 
C 12 (p,pn)C n for the reaction 6 C 12 + 1H 1 -» 6 C n + {H. 1 + on 1 . 

What physical properties of the system remain unchanged throughout 
the reaction? These conserved properties provide very valuable infor- 
mation for every reaction. We list conserved properties : 

1. In no observed process does the total electric charge change. A 
proton can change into a neutron, but a positron or positive meson must 
appear, or an electron disappear, in the process. The familiar creation 
of electron-positron pairs is a dramatic confirmation of this principle. 

2. To this date (1952) no process has been observed in which the 
total number of nucleons, i.e., the total value of A, is different on the 
two sides of the reaction. Presumably this is not a fundamental property 
of nucleons; the creation of positive and negative "proton" pairs is 
expected when sufficient energy is available. No such process has yet 
been observed. Nucleon number is conserved in all known reactions. 

3. The constants of motion of classical mechanics are conserved, at 
least to an order of accuracy beyond experimental interest. These are 
total energy, momentum, and angular momentum. The energies of 
chemical binding, and the accompanying forces and torques, are negligi- 
ble compared to the energy transfers in typical reactions; so the colliding 
particles can be regarded as a mechanically isolated system. (For some 
special reactions involving very slow neutrons, the chemical binding and 
the thermal motion of the target nuclei must be considered. Even the 
whole of a macroscopic crystal may be taken as the target in some situa- 
tions. Compare Part VII, Section 5.) The conservation of total 
energy must of course include the energy equivalent of mass changes 
of the reactants, for this will generally be a large fraction of the available 
kinetic energy. The conservation of momentum guarantees that the 
incident and product nuclei move in a plane if only two product nuclei 
result. The total angular momentum includes of course the intrinsic 
or spin angular momentum of the reactants combined with the orbital 



Sec. 1A] The Conservation Laws 3 

angular momentum of their relative motion. It is this total which must 
be conserved, though the breakdown of the total into intrinsic and orbital 
angular momenta may not be predicted in general. The experimental 
determination of the angular momentum of individual particles is nearly 
out of the question, but the statistical angular distribution of reaction 
products depends on the angular momentum relations. Measurements 
of such distributions are important aids to analysis of nuclear reactions. 

4. The "constants of motion" peculiar to quantum mechanics are 
also conserved. These are two in number: the parity and the statistics 
(S3). 1 The parity refers of course to the behavior of the wave function 
of the system upon inversion of all coordinates in the origin, i.e., upon 
the changing of right- to left-handed axis systems. No physical property 
of an isolated system can depend upon the kind of axes used, but the 
wave function can change sign under such a transformation without 
affecting any expectation values. Whether the wave function changes 
sign or remains the same under such an axis inversion is indicated by 
the parity: even for wave functions remaining unchanged, odd for those 
changing sign. The parity of the initial wave function of the whole 
system must be conserved. 

The statistics of the system is related to these constants. The inter- 
change of all coordinates of identical particles in any system can lead 
only to change of sign or to an unchanged wave function. All particles 
must belong either to the class whose wave functions change sign when 
identical particles are interchanged (these are called Fermi-Dirac 
particles) or to the class whose wave functions must remain unchanged 
(these are called Bose-Einstein particles). From the relativistic theory 
of wave fields, it can be shown that particles with half-integer intrinsic 
angular momentum are Fermi particles; those with integer spin, Bose 
particles. In ordinary reactions, where the nucleons are simply rear- 
ranged, this conservation law leads to no new conclusions; but in proc- 
esses where new types of particles — mesons, neutrinos — are created or 
destroyed, this condition sharply limits the processes which can take 
place. 

A. Application of Energy-Momentum Conservation 

1. Elastic Collisions. The simplest application of these principles is 
to the elastic collision of two free particles. The slowing down of neu- 
trons by elastic collision is the best-known example (see Part VII, 
Section 4), but, because the treatment is the basis for more complicated 
processes, we shall give the theory here, using the less familiar, but 

1 References for the text of Part VI will be found on pages 189 to 192. 



A Survey of Nuclear Reactions 



[Pt. VI 



increasingly useful, relativistic form (B7). The non-relativistic formulas 
are of course contained in the low-energy limit. 

We begin with the target particle at rest in the laboratory coordinate 
system C. It is a simplification to make a transformation to a moving 
system C in which the laboratory and target particle are of course mov- 
ing, but the total momentum of the colliding particles is zero. Lorentz 
transformation from this system to C will give us the results we want; 
we know that the energy-momentum conservation is automatically 
satisfied in any Lorentz frame if it holds in C". 

Let the target particle of rest mass m 2 be stationary in C, and the 
other particle of mass mi be incident with velocity v along the x axis. 
In the system C" the particles approach each other with velocities «/ 
and u 2 respectively. We require the particles to have the equal and 
oppositely directed momenta p' before collision. Then 



V = 



miUi 



m 2 u 2 



(1-fr' 2 )^ (1-/J 2 ' 2 ) M 
where j3' = u'/c. Using the notation 7' == 1/(1 — /3' 2 ) M , we have 



(la) 



Squaring, 



c 

«i 8 W 2 - 1) = m 2 2 (y 2 ' 2 - 1) 



(lb) 



(2) 



Since the total momentum remains zero after the collision, the final 
momenta are easily seen to be equal and opposite, but the particles 
now are receding instead of approaching. Compare Fig. 1. If particle 1 




Fig. 1. Momentum relations for collision in center-of-mass system. 

is scattered through the angle 6' in C", particle 2 will come off at an angle 
w — 6' with respect to the initial direction of particle 1. The velocities 
are not changed because the kinetic energies are conserved in an elastic 
collision. 



Sec. 1A] The Conservation Laws 5 

We now make a simple Lorentz transformation to the laboratory rest 
system C. If particle 2 is initially at rest in C, C must be moving relative 
to C with velocity u 2 in the direction of the incident motion. Now we 
know that a four-vector is formed by the momentum components and 
the total energy E/c 2 , transforming under Lorentz transformation like 
the coordinates x, y, z, t. Therefore the Lorentz transformation gives 
for the initial momentum of the first particle in the system C" (omitting 
a factor of m\C throughout) : 

(7/ 2 - 1) K = 7 2 '[(7 2 - 1) K ~ /Vy] = 7 2 '(7 2 - 1)* - (7 2 ' 2 - 1) M 7 

(3) 

where m,ic(y 2 — 1)^ is the momentum, and miC 2 y the total energy of 
particle 1 in the laboratory system C. The kinetic energy is m\C 2 (y — 1). 
Using (2) and (3), we find 

■ -2/ 

Ti = 7 : , .^u (4a) 



72 = 

1 + 




(4b) 



To obtain relations between the scattering angles, we again use the 
fact that the momentum and energy E, divided by c 2 , form a four-vector. 
The total energy E 2 of particle 2 after the collision and its momentum 
components, in the laboratory system, are given by 

E 2 = 72 W - c/SaV cos 6') (5a) 

Px 2 = 7 2 y (-p' cos 0' + - /9 2 'tf 2 'j (5b) 

P»2 = -P'sinfl' (5c) 

where the incident direction is the x axis, and all the momenta lie in a 
plane. The energy of particle 2 in C is E 2 ' = m 2 c 2 y 2 ', and its momen- 
tum p' = m 2 y 2 'c(3 2 '- Dividing (5b) by (5c), we find the angle of scatter- 
ing $2 of particle 2 in the laboratory system C: 

1 1 

tan 9 2 = -cot - 6' (6) 

72' 2 



6 ' A Survey of Nuclear Reactions [Pt. VI 

The minus sign means that, if particle 1 is scattered above the x axis 
into the region of positive y, particle 2 recoils below the axis. If 0' is 
180° in the center-of-mass system C, 6 2 = and particle 2 recoils for- 
ward with maximum energy. As 0' decreases toward zero, particle 2 
recoils with decreasing kinetic energy at angles approaching 90°. The 
case 0' = corresponds to grazing collision, in which particle 1 loses no 
energy and particle 2 recoils with zero velocity at 90°. 

The recoil kinetic energy of particle 2 in the laboratory system is 
7' 2 = E 2 — m 2 c 2 . From (5a) 

T 2 = m 2 c 2 (y 2 ' 2 - 1)(1 - cos d') (7a) 

Using (6) and (4b), we can write this in terms of the angle d 2 . Finally 
we may introduce the total energy E = m,\<?y and momentum p = 
mic(y 2 — 1)^ of the incident particle, getting 

p 2 COS 2 2 

T 2 = 2m 2 c 2 (7b) 

(E/c + m 2 c) J — p 2 cos 2 2 

Evidently 2' 2 (max) = 7' 2 (0 2 = 0). 

It is of interest to compute the maximum fraction of the kinetic 
energy of the incident particle which can be transferred. T = E — m\C 2 
= mic 2 (T — 1), and 

7V max) _ 2( 7 + \)m x m 2 _ 2( 7 + \)m x m 2 

T m 2 + 2ymxm 2 + m 2 2 2(7 + l)mim 2 + {mi — m 2 ) 2 

This fraction increases monotonically with incident energy from the 
value familiar in the non-relativistic limit, 4m 1 m 2 /(mi + m 2 ) 2 , to unity 
for very high energies, with 7 » 1. If the particles have equal mass, 
the maximum recoil energy is of course always equal to the incident 
energy. 

If the particle incident is light (e.g., electron or photon), particle 2 
can acquire relativistic energies only if p ■= m 2 c 2 or 7 = m 2 /m\. This 
is also the condition for transfer of a large fraction of energy from par- 
ticle 1 to particle 2, if a heavy particle is incident on a light target particle. 

We now discuss the relation between the scattering angle 0i of particle 
1 and its energy loss. The total energy of particle 1 in the laboratory 
system C corresponding to a scattering through 0' in C is 

E x = m lC 2 [7i'72' + (7i' a - 1) } ^(72' 2 - 1) H cos 0'] (9) 

where the 7"s are as given by (4a) and (4b). Now we need cos 0' in 
terms of the angle measured in the laboratory system. For this we 



Sec. 1A] The Conservation Laws 7 

apply the Lorentz transformation to the energy and momentum of 
particle 1 in the manner of (3). This shows 

1 sin 0' 

tan 0! = (10) 

y 2 (cos 0' + TO 1 Y 1 '/W12Y2') 

Solving for cos 0' in terms of tan 0\, we get a quadratic equation with 
two roots, indicated by the ± sign: 



cos 0' = 



mi , 
7l ' T2 ' tan' 5 0x 



(1 + T2' 2 tan 2 X ) L m 2 

2 ™ 2 



1 5 tan 2 ^) 



(1-1) 



Since the energy for a definite scattering angle in the laboratory system 
is given by (9) in terms of cos 0', it appears that there are two possible 
values for the energy of a particle elastically scattered through a definite 
angle. It is instructive to examine the situation graphically. 

For non-relativistic velocities (7 = 1), Eq. (11) gives the tangent 
of the scattering angle in C in terms of functions of the angle 0' in C". 
For higher velocities, we can define an auxiliary angle ^ by the relation 
tan \p = y 2 tan0j. This angle \f/ can be obtained geometrically by 
adding a vector of length m^ix /m 2 y 2 along the x axis to a unit vector 
whose components are cos 0' and sin 0'. The center of a unit circle 
represents C", and the scattered particle may go off in any direction, 
along a unit radius vector. The auxiliary angle ^ is then just the angle 
between the x axis and the resultant of the radius vector in C added to 
the vector miyi'/m 2 y 2 '. The construction for the three cases we must 
consider is shown in Fig. 2. 

In case (a), with mi > m 2 , there are two possible directions of scatter- 
ing in C" (i.e., two distinct orientations of the unit vector) for one scatter- 
ing angle 0i in C. There are thus two values for the energy of particle 1 , 
in agreement with (11). 

For the case mi < m 2 , Fig. 2b is appropriate. Here the a>axis vector 
is less than unity, and the construction gives only a single angle 0' to 
be associated with a definite i/'. The analogue to the second orientation 
of the unit vector in case (a) corresponds to the angle (x — \f/) and gives 
no additional solution. One of the roots of (11) is to be rejected if 
mi < m 2 . From the diagram we can see that, if 0i is less than 90°, we 
should take the more positive value of cos 0' in (11); for X greater than 
90°, the more negative one. 



8 



A Survey of Nuclear Reactions 



[Pt. VI 



We shall treat case (c), mi = m 2 , as a limit of either (a) or (b). From 
the figure we can see that there is an upper limit to ^ and thus to the 
angle 1; corresponding to the case where the line defining \p is tangent 



(a) m 1 >m z 




Unit circle 



(b) m 1 <m 2 



(c) m^Mj 




Fig. 2. Scattering angle relations for elastic collisions. Incident particle is mi; 
target particle, m^. The angle 41 determines the laboratory scattering angle. See 

text for construction. 

to the unit circle, and angles 0'(u, 0\ 2 ) have coalesced to a single 
value. The radical in (11) must vanish at this maximum value: 



tan^""" = 



m 2 



m\ — m 2 



(12) 



9i (max) = arc s i n 



m 2 
nil 



In case (b), with m\ < m 2 , scattering can clearly occur for all angles 
between and t. In the limiting case (c), mi = m 2 , the condition of 
tangency is reached for ir/2. 

In the special case (c), all the formulas above are much simplified. 
From (4a) and (4b) we get 

Ti' = 72' = {—£-) ( 13 ) 



Sec. 1A] The Conservation Laws 9 



If T* is the kinetic energy of the particle scattered through angle 6 in C, 
either (5a) or (7b) gives 

2(7 - 1) cos 2 

(7 + 1) ~ (7 - 1) cos 2 
and, with T = mc 2 (y — 1), 

T cos 2 e 



T* = mc 2 — (13a) 



1 + f (7Vmc 2 ) sin 2 



mi — m 2 = m (13b) 



Although there are still two roots of (11), one of them always corresponds 
to 0' = 180°. This means of course that one of the colliding particles 
is projected in the forward direction with the full incident energy, while 
the other remains at rest. The second root of (11) gives (13a). 

The familiar non-relativistic form of (11) is easy to obtain. If 
»ti = m 2 , the scattered particles always come off at right angles to each 
other, with energies proportional to the squares of the cosines of the 
angles of deflection. For higher velocities, however, the angle included 
between the directions of the outgoing particles is always less than a 
right angle. Changing the sign of 2 in (6) and using (10) and (13a) 

and (13b), we find 

2 

tan di tan 2 = (14) 

1 + 7 

Since y is always greater than unity, 0i + 2 is less than 90° except for 
the grazing collisions in which particle 2 may be thought of as scattered 
at 90° with zero velocity. A sufficiently accurate measurement of 
angles 0i and 2 would serve to determine the incident energy. This 
method is more sensitive for large values of y. 

Finally we set out the familiar relations for the non-relativistic case. 
If T 2 is the kinetic energy of recoil of particle 2 and T is the incident 
kinetic energy of particle 1, 

T 2 =- -^cos 2 d 2 T (15) 

(mi + m 2 ) i 

For the final kinetic energy of particle 1, from the proper limit in (9), 
we obtain 



\mi + m 2 / L \m\/ 

/m 2 2 . y 
± 2 cos 0i I — r — sin 2 0! I 



The treatment of the two roots is the same as in Fig. 2. 



(16) 



10 A Survey of Nuclear Reactions [Pt. VI 

2. Collisions with Creation of New Particles. For sufficiently high- 
energy collisions of nuclear particles, not only are rearrangements of 
nucleons possible, in which kinetic energy is transformed into binding 
energy (or the reverse), but also new particles may be produced whose 
rest energy comes from the kinetic energy of collision. In a collision in 
which additional particles are produced whose rest energy totals Mc 2 , it 
is clear that energy conservation requires that at least the incident kinetic 
energy exceed Mc 2 . In addition, because the conservation of momentum 
requires that the center of mass of the whole system move with un- 
changed momentum, we must usually provide additional energy to fulfill 
this condition. 

Let us consider the threshold value of the kinetic energy of the incident 
particle, ra x , on a stationary target particle m 2 (F8). The threshold 
is the minimum energy below which the production cannot take place. 
How probable the production becomes just above threshold is of course 
impossible to tell in general; it will depend upon the particular inter- 
actions being considered. It is easy to see that at the energy of threshold 
the total energy of all particles is a minimum after collision. In the 
frame C" we know that the final momentum must be zero. These two 
conditions can be satisfied if all particles after collision are individually 
at rest in the center-of-mass system C". Then the initial kinetic energy 
must have been just equal to the increment in rest mass energy, Mc 2 . 
The total energy in the C" system before collision is 

E t ' + E 2 ' = mi c 2 yi ' + m 2 c 2 y 2 ' 
and from (4) we can write this 

Ex' + E 2 ' = c 2 (w! 2 + 2ym } m 2 + m. 2 2 )' A (17) 

Equating it to the energy after collision, 

Ei' + E 2 ' = c 2 ( mi + m 2 + M) 

Solving (17) for y, we obtain the threshold kinetic energy T of mi in the 
laboratory system from the relation T = m A c 2 (y — 1). This gives 

Mc 2 {m l + m 2 + IM) 

T = LJ *—L (18) 

m 2 

To create a nucleon pair from a proton-proton collision, for example, 
would require T at least 2 ■ m p c 2 (3m p /m p ) = 6m p c 2 . If the pair were 
made by a gamma-ray or electron incident {m x ^ 0), the threshold 
would be T = 4m p c 2 . 

In such collisions the target particle need not be at rest. Here we 
may think of a target nucleon as one of the nucleons bound in a nucleus, 



Sec. 1A] The Conservation Laws 11 

and thus moving with the velocity characteristic of its zero point vibra- 
tions. If the collision occurs when the target nucleon is moving with a 
velocity component directed toward the incident particle, less incident 
beam energy will be needed. If the target particle m 2 has the initial 
velocity v 2 in the x direction, Lorentz transformation shows that the 
threshold energy of (18) becomes reduced to the value 

T min = y*(T - v 2 p) + m lC 2 ( T2 - 1) (19a) 

where p is the initial momentum of particle mi, and 72 = (1 — v 2 2 / c 2 )~ V2 . 
A further reduction comes about if the collision occurs with capture of 
the incident particle so that binding energy can be released. 

It is of some interest to discuss the possible angular and energy dis- 
tribution of the produced particles. If we restrict ourselves to the case 
in which the target particle is at rest, and in which only one new particle 
of mass M is made, we can draw some simple conclusions. It is clear 
that at the threshold energy there is only one final condition: all the 
particles move forward with the same velocity, since they are all at rest 
in the center-of-mass system. As the energy exceeds the threshold, the 
created particle will in general be able to travel in a distribution of 
angles around the forward direction. The details will of course depend 
on the nature of the interaction. But some limits can be given in general. 
It is clear that the maximum momentum of the new particle will corre- 
spond to a case in which M is traveling forward and the two original 
particles move opposite to M in the center-of-mass system, with the 
same speed for each. This leads to a value for the maximum kinetic 
energy of M in the laboratory system : 

Mc 2 + r m « = 

\c 2 [{ynii + m 2 )A + mi(y 2 - \) }i B]/{mi 2 + 2ym 1 m 2 + m 2 2 ) (19b) 

with A = M 2 + 2(t - \)m x m 2 and B = {[2(7 - l)m 1 m 2 - M 2 ] 2 - 
AM 2 (nix + %) 2 ! ^. This has a very simple form at the threshold: 

Mc 2 Mini! + M/2) 

T max = — (20) 

m 2 mi + m 2 + M 

The angular distributions will be controlled by the relation between 
the maximum velocity with which M can move off in the center-of-mass 
system and the velocity of C with respect to C". Arguments similar to 
those accompanying Fig. 2 show that here too there are two possibilities : 
the new particle will be emitted only in part of the forward hemisphere 
if its maximum center-of-mass velocity vm = c 2 pm'/Em' is less than 
the velocity of the center-of-mass system, c/3 2 '. Then the situation is 



12 



A Survey of Nuclear Reactions 



[Pt. VI 



like Fig. 2a, and a minimum energy for M in the laboratory exists; it is 
given by (19) with a minus sign instead of a plus sign before the radical. 
If v M ' > c/V, however, the case is like that, of Fig. 2b, and all angles of 
emission are allowed in the laboratory frame. The minimum energy for 
M is then zero in the laboratory system. 

3. Reaction Energies. The most studied nuclear reactions up to the 
present time involve neither elastic collisions nor the production of new 
particles. They consist of the rearrangement of nucleons, with the 
ejection of different nuclei having correspondingly changed binding 
energies and rest masses. Just as in chemistry, the heat of reaction Q is 
a significant quantity. The energy Q is the energy released as kinetic 
energy (or the energy of photons, etc., set free) at the expense of the 
internal energy of the colliding systems. We may define Q by either 
of the relations: 

Q = c 2 ( £ tru -EmA = -I Ti + £ T f (21) 

V initial final / initial final 

where the notation is that of Section 1A2, and the equivalence of the 
definitions follows from the conservation of total energy; T { is defined 
as kinetic energy. We consider first the case of a typical reaction of the 
two-product type: T(i,p)R. 

It is very easy to treat this case with the target nucleus at rest. Then 
conservation of momentum in the laboratory system yields the vector 
relation P; = p p + Pr. We have only to square this, employ the 
relativistic connection between momentum and kinetic energy T, 
p 2 = T 2 /c 2 + 2mT, and substitute in the definition for Q, eliminating, 
say, T R . We have 

\ 2 T 



Q = T P -Ti + m R c 2 



+ 



m R / m, 



X 



T T 



IrripC ■ m-iC 



/m p V T p / [ T p \ 
\tor/ m p c 2 \ m p c 2 / 

-(— +2)- 

{ C 2 \TOjC 2 / 

m p &/ \ rriicr/ 



m p c 

2m p nii 
ra R 2 



- 1 



(22a) 



This may be written in the low-energy limit by expanding the radical, 
neglecting T/mc 2 compared to unity for all particles. The familiar 
result follows: 



m / m v\ ( w,- \ (m v T v mi'. 

Q = T p [l+—)-Ti[l -)-2 P P 

\ TOr/ \ Tor/ Tor 



miTi)> 



Vz 



cos 6 P (22b) 



Sec. IB] The Conservation Laws 13 

In reactions of this type, with only two product particles, there is evi- 
dently a unique value of T p for a given recoil direction and a given inci- 
dent energy. When the reaction produces three or more particles, no 
such simple connection exists. The case is then parallel to the discussion 
in Section 1A2. A distribution of energies results. If the reaction takes 
• place in two independent stages, first the emission of two particles, then 
the break-up of one of the pair, somewhat simpler relations can be 
obtained. Thus, study of the energy distribution can give some insight 
into the mechanism of the reaction. 

The rigid correlation between energy of outgoing particle and angle of 
emission has been much exploited as a laboratory means for obtaining 
particles of a well-defined energy. For neutrons especially such mono- 
energetic beams can hardly be obtained in any other way. Simply as 
an example, it is interesting to note that the very exothermic reaction 
T 3 (d,w)He 4 used with a well-controlled 1-Mev beam of deuterons pro- 
duces neutrons ranging from almost 16 Mev to under 13 Mev in the 
backwards direction. A study of a set of such reactions has been made 
which makes possible the production of monoenergetic neutron beams of 
energies from a few kilovolts up to about 20 Mev; a source of charged 
particles of well-defined but rather modest energies of a couple of Mev 
was used. A valuable review by Hanson et al. (H2) fully discusses the 
several reactions used most frequently and gives a graphical treatment 
of the fundamental relations. Graphical treatments of the relativistic 
case, very useful when it is desired to reduce experimental data, have 
been given in (B16) and especially in (Ml). 

It is worth while to make the remark in closing that, in all the relati- 
vistic formulas above, the case in which some particle (i) is a photon can 
be obtained by replacing the total energy Ei = mic 2 y by the quantum 
energy hv, and neglecting m,- wherever it occurs without a factor 7. 

B. Conservation of Angular Momenta 

The complementary relation in quantum mechanics between energy 
momentum and space-time leads to straightforward experimental use 
of the ideas of energy and momentum, as we have seen. The important 
canonical variable, angular momentum, is conjugate to angle. We 
measure the angular momenta of quantum systems mainly by obtaining 
statistical distributions in angle. The typical nuclear reaction is carried 
out with a beam of particles ; the very term beam implies a more or less 
sharp definition of the direction of motion, and hence a necessarily 
rough limitation on angular momentum. An infinite plane wave, indeed, 
contains all angular momenta of orbital motion, with definite ampli- 
tudes, and in addition requires supplementing by the wave function 



14 A Survey of Nuclear Reactions [Pt. VI 

factor representing the intrinsic angular momentum of the particles in 
the beam. Only when there is some reason to limit the states of angular 
momentum which are of interest — either simply formally in some 
systematic counting of all angular momentum states (as in the method 
of partial waves) or physically because for some reason only a limited 
number of angular momenta contribute to a given reaction — can the 
angular momentum conservation law be of much value. For example, 
in the familiar case of thermal neutron reactions, where the incident 
wavelength is large compared to the region of possible interaction, only 
the spherically symmetric component — angular momentum equal to 
zero — of the incident wave can possibly contribute. Other parts of the 
wave are of vanishingly small amplitude; the "centrifugal barrier" 
keeps them from the region of interaction. If any specification can be 
made of the angular momentum (hence of the variation with direction) 
of the incoming wave, general rules may often become helpful. 

The total angular momentum is certain to be conserved; the forces 
between nucleons are, however, non-central, so that there is a tendency 
to interconvert intrinsic or spin angular momentum with that associated 
with orbital motion. Only in special cases can the relative apportioning 
and hence the complete angular distribution be obtained (Y2). The 
spin angular momentum affects, of course, not the direction of motion 
in space but the orientation of the spin axis of the particle concerned. 
If an unknown spin change can occur, this will clearly affect predictions 
of angular distribution in a way calculable only under further specifica- 
tion of the interactions involved. There are three general results, inter- 
esting to present, which apply quite generally to all nuclear reactions 
between unpolarized target and unpolarized incident particles: 

1. As is clear on physical grounds, there will be axial symmetry about 
the direction of the incident beam. 

2. If, among the incoming partial waves, only those of angular mo- 
mentum L or below contribute appreciably to a reaction, the angular 
distribution of any single product particle cannot be more complicated 
than that of the incoming contributing partial wave. Indeed, the angu- 
lar distribution of the outgoing particle will be a polynomial in cos 6, 
where 6 is the polar angle of emission relative to the beam direction, of 
degree no higher than 2L. This holds independently of the spin of any 
particles or of the number which take part. It is restricted to non- 
relativistic velocities for the incoming particle. In relativistic cases, 
one higher power of L may occur, but in general it will be reduced by a 
factor of the order of (v/c). If several partial waves of different L con- 
tribute, the polynomial will generally contain all powers of cos 6 from 



Sec. 2A] The Data of Nuclear Reactions 15 

the maximum down; if only one wave contributes, only even powers of 
cos 6 can occur. These results refer, of course, to the center-of-mass 
frame of reference. 

3. If the distribution of product particles contains any odd powers in 
cos 8, i.e., if it is not symmetrical with respect to the plane normal to the 
beam, two (or more) intermediate states are involved with opposite 
parities. Thus an even polynomial in cos 9 is a typical consequence of 
reactions involving a marked resonance; this will be discussed more 
fully later. 

Most frequently such considerations yield useful information only 
when there is additional information, for example data permitting the 
original specification of what partial waves can contribute. This implies 
some statements about the mechanism of the reaction, some definite 
nuclear model, and some assumption about the character of nuclear 
states and of interactions. Then the selection rules which result are 
often powerful discriminants between possible alternatives. Such cases 
occur most frequently in nuclei lighter than neon, in which region some 
progress has been made in constructing detailed models for nuclear 
systems. Examples will be given in the proper places later. 

SECTION 2. THE DATA OF NUCLEAR REACTIONS 

The conservation laws are satisfied in every single nuclear collision. 
But, besides data on the constants of motion, nuclear physics also is 
concerned with knowing how many nuclear processes of a definite kind 
take place under given conditions. It is evidently beyond us to say 
whether or not the next proton, say, will initiate a transmutation in a 
given fluorine nucleus, but we can obtain from both experiment and 
theory the probability of the process. We can expect to predict with 
calculable accuracy the fraction of the beam of incident protons which 
enters the target nuclei. 

A. Yields and Cross Sections 

The simplest expression for the probability of a nuclear reaction is 
the yield. This is frequently but a semi-empirical expression, stipulating, 
for a particular experimental arrangement, how many processes occur 
per incident particle. The statement that a yield of 1/2000 was obtained 
for the reaction Li 7 (p,a)a with a thick target at 3 Mev means only that 
one lithium nucleus divides for every 2000 protons stopped in the target 
material. The specification is obviously incomplete. For data about 
the probability of the reaction itself one would need to know how many 



16 A Survey of Nuclear Reactions [Pt. VI 

encounters occurred and at what energies, whereas here the protons lose 
energy as they penetrate the target, which is often not even a single 
nuclear species but a mixture of isotopes or even a compound. Improve- 
ment could be obtained by using a target so thin that the energy loss of 
the beam protons in traversing it was negligible ; here at least the energy 
would be sharply defined. Thin-target or thick-target yield data are 
common in the literature; each requires special interpretation before 
any absolute number can be obtained (CO, Rl, R2). Such a form of 
presenting data implies that only some relative feature of the reaction 
is under study. 

The familiar cross section is a complete specification of the probability 
of nuclear reaction. A cross section oab for a reaction in which the par- 
ticle A is incident with specified properties and the specified set of 
particles B emerges can be defined by the expression n# = Ia^abN. 
Here N is the number of target particles presented to the beam (for 
example, for thin targets the product of beam area, target thickness, and 
number of target atoms per cubic centimeter gives JV), n# is the number 
of the specified particles B emitted per second, and I a is the incident 
flux of particles A, the number per unit area and time. 1 

A cross section may be specified for any process or partial process: 
for example, we can assign a cross section for the emission of gamma-rays 
of energy E in the direction specified by the polar angles (8, <p) when 
protons of energy W are incident on Li 7 . Then the number of such 
gamma-rays emitted in that direction per unit solid angle and per unit 
energy for unit proton flux is just <r py (E, 6, <p) for each lithium nucleus 
in the beam. The quantity a P1 is called a differential cross section (per 
unit energy and unit solid angle), since it is defined for an infinitesimal 
range of the continuous variables E, <p, and 8. The total number of 
gamma-rays emitted in all directions is of course obtained by summing 
over all energies and directions. Then the number of gamma-quanta 
for unit flux and a single target atom is 

S P7 = / dE I dip I d(cos 8)<r py (E, <p, 8) 
Jo Jo J +1 

Here 2 PT would be called the total cross section for proton capture by 
lithium for protons of energy W. In this way the frequency of any 
process or partial process can be described fully by giving the appropriate 
cross section, which may depend on any set of discrete and continuous 

1 Note that for the collisions of identical particles, as in the scattering of protons 
on protons, incident and recoil particles are not physically distinguishable, and the 
number of observed events may then be a sum of both. 



Sec. 2B] The Data of Nuclear Reactions 17 

variables. The cross section for any process is just the area of the inci- 
dent beam from which particles are removed by the given process, when 
the beam is thought of as directed at a single target nucleus. Of course, 
the cross section may be denned not only per nucleus, as we have here 
done, but also per unit volume, per gram, or in terms of any convenient 
measure of target atoms. If there were no non-classical wave effects, 
the cross section for a definite target, summed over all the possible 
processes which could occur, would be just the geometrical area of the 
target. We shall see almost this result for the removal of fast neutrons 
from a beam (B14, L2, M3, W5). 

The probability of a nuclear reaction, expressed either as a cross 
section or only roughly as a relative yield, is most frequently measured 
for a specific nuclear reaction as a function of the energy of the incident 
beam. Such a number is called the excitation function of the reaction 
concerned. (If a change in chemical identity is involved, it is sometimes 
called a transmutation function.) The chief data of nuclear experiments 
besides the energy relations and the nature of the reaction are the angular 
distributions and the excitation functions, which can be reduced to a 
knowledge of the cross section for the specified reaction as a function of 
energy of incidence and direction of emission. 

B. The Measurement of Yields and Cross Sections 

The experimental procedures for the measurement of cross section 
amount to a good fraction of the subject matter of experimental nuclear 
physics. It would be presumptuous to try to discuss them in a small 
space. It seems worth while, however, to give a kind of summarized 
functional enumeration, not so much of the experimental methods, but 
of the general procedures and precautions which must be included in any 
program for the measurement of cross sections. Evidently the type of 
measurements made will depend on the detail and the accuracy of cross- 
section knowledge wanted; it is obviously more exacting in general to 
seek a knowledge of the differential cross section than of the total cross 
section, and of an accurate absolute value than of a rough relative one. 

1. The Beam. If the cross section a-,-, t(E, 6,<p) is sought as afunction 
of energy, of angle, of type of projectile, it is obvious that the beam 
must be well defined with respect to these quantities. Usually this 
definition is imposed on the entire incoming beam of projectiles; some- 
times it is possible to study the reaction products and by some condition 
of the reaction eliminate all events which did not begin with the correct 
value of one of the variables. These methods frequently involve time- 
coincidence counting. Generally the entire beam accepted into the re- 
gion of reaction is known to have the wanted values of its parameters. 



18 A Survey of Nuclear Reactions [Pt. VI 

The type of projectile is the most general variable which needs 
control. The emerging beam from a cyclotron is likely to be quite free 
of all types of ions other than the resonant species, but an electrostatic 
generator, for example, will equally well accelerate the deuterons and 
the protons which may leave its hydrogen-ion source. Molecular ions 
contribute to the current like atomic ions, but not to the yield, and 
must often be specifically excluded. For such purposes magnetic resolu- 
tion of the raw particle beam is often used. When unstable particles 
form the beam, as in the case of pi-mesons, any stretch of beam travel 
will allow decay particles to enter the beam. Such beams cannot be 
wholly pure, and corrections for the mixture of particles are needed. 
Usually charged particle beams are rather easily controlled for beam 
purity because of the power of ion-optical analysis. Gamma-ray beams, 
or beams of charged particles energetic enough to yield neutron-emitting 
reactions in any intervening windows, air, etc., in the path, may require 
special care. Crude magnetic analysis will generally remove electrons; 
neutrons are much harder to eliminate and may cause trouble. Deflec- 
tion of the wanted charged beam will in principle solve the neutron 
problem, if scattering from the magnets, etc., is not too severe. The 
problem is one of increasing importance as beam energy, and with it 
the number of secondary reactions, increases. 

The energy is often very well defined by the source used. Control of 
d-c ripple in electrostatic generators has reached precisions of a few 
parts in 10 s . Magnetic analysis can be pushed nearly as far even for 
roughly defined beams like the cyclotron output. Synchrotrons give 
excellent control over beam energy as well. Any matter which the beam 
traverses in its path to the target produces energy losses by ionization 
and excitation of atoms, and also by nuclear events, which smear out 
the beam energy. This is in principle avoidable by keeping the beam in 
vacuum, but eventually the beam must strike some target material. 
The same processes, of course, occur in the target, which must be kept 
thin (measured in atoms per square centimeter) if the cross section is 
wanted at a well-defined energy. Thick targets can yield only cross 
sections integrated over energy, and may cause straggling in direction 
and even particle type as well. Yield as a Junction of energy sometimes 
can roughly replace beam homogeneity by allowing a differentiation of 
the integral yield. 

Homogeneity in energy usually is easier to secure than absolute 
knowledge of the energy of the beam employed. The energy scale in 
the region of Mev depends on comparison with a few nuclear resonance 
energies which have been calibrated absolutely by measuring the trajec- 



Sec. 2B] The Data of Nuclear Reactions 19 

tory of a proton beam in a known electric field. 1 This gives an accuracy 
of about one part in a thousand. Magnetic trajectories using the proton 
magnetic moment resonance frequency for field measurements work 
about as well, and extend into the 100-Mev region, especially for elec- 
trons from gamma-ray beams. Range measurements using the theory 
for energy loss by ionization are very valuable secondary standards over 
all energies above, say, a few kev. They have been computed and 
calibrated over this whole range for nearly all the usual projectiles. 
The angle of emission of the Cerenkov electromagnetic shock wave in 
a material of known index of refraction is an elegant method for particles 
with a velocity such that j3n > 1, where n is the refractive index of the 
material; velocity measurements by direct time-of -flight determination 
have also been made. The kinematics of elastic collisions is also capable 
of giving beam energies expressed as a ratio to the rest energy of the 
particle, in the relativistic region. 

Angle definition for the beam, often called collimation, is of impor- 
tance for any measurements of dcr/dQ. Here defining slits of some sort 
are the usual key parts of the apparatus. Sometimes the direction of 
the track of the product particles themselves, made visible by cloud 
chamber or nuclear emulsion techniques, replaces slits followed by 
direction-insensitive counters. The effects of Coulomb scattering from 
nuclei and electrons of the target itself or any material in the beam path 
are the main enemy of good collimation. Divergence of the original 
beam forms in itself no source of error which collimating slits or simple 
distance of beam travel and choice of detector acceptance angle cannot 
remove. But good collimation implies loss of total beam intensity, and 
practical limits of time and background here often are decisive. 

Perhaps the most important of all parameters of the beam is the 
number of particles striking the target in a specified experiment. Often 
the current is measured and then time-integrated. Monitoring, or con- 
trol of relative intensities, is frequently used for a series of experiments 
in which absolute values are either ignored (e.g., all cross sections may 
be measured relative to a given known cross section) or calibrated 
against one single measurement. Integrated charge may be measured 

1 Typical reference energies in absolute volts are 



F 19 (p,a T )0 16 


0.3404 Mev 


Resonance 


Li 7 ( P)T )Be 8 


0.442 


Resonance 


F 19 (p,q:t)0 16 


0.8735 


Resonance 


AF(p, T )Si 28 


0.9933 


Resonance 


Li 7 (p,n)Be 7 


1.882 


Threshold 



See, for example, W. Hornyak, T. Lauritsen, P. Morrison, and W. A. Fowler, Revs. 
Modern Phys., 22, 291 (1950). 



20 A Survey of Nuclear Reactions [Pt. VI 

by measuring the potential change or integrated current flow to a Fara- 
day cup, an insulated conductor so arranged that no secondary charges 
can leave it and all beam charge is stopped within it. In one high-energy 
proton experiment the Faraday cup was a forty-pound chunk of brass 
under magnetic field in a vacuum. More usually it is a small cylinder 
with a beam window in one end, Ionization is often used to measure 
intensity by taking advantage of the known space rate of energy loss 
and the mean energy used per ion pair. Direct counting of ionizing 
tracks in photoplates can be used. Relative values can be obtained by 
actually measuring the beam-induced yield of some known reaction, 
using radioactivity, particle emission, or even collecting the neutral 
atoms of a stable reaction product, as has been done for such reactions 
as IA 6 (n,H s )a. Even calorimetry of a target which stops a totally 
absorbed beam has been used for beam intensity. 

With particles of non-zero spin (protons or even gammas) there is 
another possible parameter to control: the; spin component. This cor- 
responds to use of a polarized light beam in optical experiments. Apart 
from magnetic polarization of neutrons, only the slight polarization 
effects resulting from double scattering have been examined for charged 
beams. This may become a more important field in the future. 

2. The Target. To specify a cross section precisely, the nuclear 
composition of the target must be well known. This implies pure and 
homogeneous targets. The isotopic concentration of the target must, 
of course, be known ; this is usually easy if natural targets are used, but 
special analysis is required with targets prepared from isotope-separated 
material. 

The target nucleus may undergo the looked-for reaction, but in gen- 
eral this is improbable. Most of the target nuclei are not changed by 
the beam. Their presence is needed because the probability of reaction 
is proportional to the number of nuclei independently exposed. But 
their very presence modifies the beam by producing straggling in energy 
and direction by Coulomb scattering from electron and nucleus, or by 
occasional nuclear collisions of kinds other than that under study. 
They may produce a "straggle" in beam particle type by yielding 
secondaries which go on in the target to induce unwanted reactions of 
nuclei later in the beam path. They clearly modify the beam intensity. 
Obviously the use of targets sufficiently thin that the changes in any of 
these properties is small compared to the resolution desired in each 
parameter is always a solution. But it may be impossible to employ 
such a target because the total number of reactions then becomes too 
small, or because the physical form of the target elements prevents 
preparation of a thin sample. Then a careful study of the beam be- 



Sec. 2B] The Data of Nuclear Reactions 21 

havior in the target material must be made, and corrections made for 
all the straggling effects. A frequently employed scheme of finding 
cross-section variation with beam energy is to use a target made of a 
stack of thin foils, the yield in each of which is measured. The energy 
variation of the beam with target thickness tells the mean energy in 
each foil, and thus the excitation function is measured. The scheme 
demands that the range of the product particles in the target material 
be less than the range intervals used in choosing the foil thickness; 
otherwise the yield function is partly displaced in a distance and energy 
scale, and detail is lost. 

Experimental problems of great importance sometimes arise from the 
heat dissipation and other physical effects of strong ionizing beams. 
Constancy of the target mass and composition is assumed in nearly all 
measurements; evaporation, diffusion, coatings from soot of decom- 
posed pump oils, mechanical displacements of target material by melt- 
ing or otherwise, and radioactive recoil must all be avoided or corrected 
for. 

3. The Measurement of Cross Sections. The most direct method of 
measurement is an initial absolute measurement of some differential 
cross section which defines experimentally the angular acceptance, 
particle type, energy, etc., of the reaction product. Then a count of 
the number of such events from a fixed target thickness (with correct- 
tions pointed out above) and with known beam population yields the 
cross section directly. The means employed are diverse. 

(a) Measuring the Residual Nucleus by Detecting Radioactivity. This 
may be connected with a radiochemical separation of that nucleus from 
the target matrix. Such a scheme does not define the energy, angle, 
spin, or any other property of the product nucleus except its chemical, 
or indirectly its nuclear, identity. It may be combined with the thin- 
layer target scheme to get excitation functions for a specified reaction. 

(b) Collecting the Product Particle. If this is done, as usual, while the 
product particle is in flight, the angle, energy, and type of product may 
all be defined. Very often ion-optical means are used. Such schemes 
separate product particles by their differing trajectories in a vacuum 
chamber with a controlled electromagnetic field and make possible 
detection of individual levels of the residual nucleus by careful meas- 
urement of the energy released. The charge-mass ratios usually fix the 
product uniquely. Detectors are used to count the product particles, 
which may be linearly responding ionization chambers or solids giving 
scintillation response, gas-amplifying counters like the proportional and 
Geiger counters, photoemulsions, cloud chambers, simple charge collec- 
tion by Faraday cup, or specific induced nuclear reactions themselves 



22 A Survey of Nuclear Reactions [Pt. VI 

(W10). The angular resolution is evidently mainly a problem of the 
disposition of the apparatus. Energy resolution may be secured by 
such ion-optical means as we have mentioned, or by using absorbers to 
fix the total energy needed before particle counting. Counter telescopes 
which require energy loss rates to be within a fixed range, determined by 
signal levels and the coincident or anti-coincident response of a fixed 
set of counters and absorbers, are very much used at the high-energy 
end, where ion optics is more difficult. Cerenkov counters and time-of- 
flight measurements are among the newest techniques coming into use 
in this field. 

It is possible to measure total cross section, a = I dil d<r/d&, more 

easily. Here what counts is removal from the beam. All that is needed 
then is a comparison of detector response with and without the inter- 
vening target. For thin targets having N atoms/cm 2 and "good geom- 
etry," then, I/I = e~ N ". "Good geometry" means such collimation 
that all particles scattered through any small angle miss the detectors 
(B14). Since the detectors have finite acceptance in every case, correc- 
tion is needed for scattering-in (P3). When this correction is sizable, 
the case is said to be one of bad geometry. Thick targets, where multi- 
ple collisions are not negligible, require additional care. The inescapable 
relation in Eq. (45) between an absorption cross section and elastic 
scattering means that special care must be taken to correct for the very 
low-angle forward coherent scattering if the geometry is very good, 
since scattering-in cannot be estimated from the angular distribution 
of the main absorptive process. Apart from such questions, it is in 
general possible to make the only absolute cross-section measurement 
one of total cross section, done by beam attenuation, and then only 
relative measurements for all the differential cross sections are needed. 
4. Special Problems with Gamma-Rays. The use of gamma-rays as 
beam projectile, or their measurement as product, has special problems 
arising from the small interaction with matter. Gamma-rays can be 
detected only after they have given their energy at least in part to mat- 
ter, especially to electrons. For this purpose most gamma-ray-detecting 
schemes make use of a layer of matter of appropriate thickness placed 
ahead of the detector proper. In this converter layer the gamma-rays 
produce electrons (or positrons) whose ionization, range, energy, and 
momentum may be measured by the means described above. At ener- 
gies in the range of a few Mev it is possible to get absolute measure- 
ments of gamma-ray flux by using the known ionization of the electrons 
secondary to gammas, which of course have a spread of energies as long 
as Compton scattering is important. At energies above 10 Mev or so, 



Sec. 2B] The Data of Nuclear Reactions 23 

a thin foil in which electron-positron pairs are the principal products 
may be used. Then a knowledge of the pair production cross section 
from theory or from total attenuation measurements may be used to 
compute the incident flux. Calorimetry is possible for strong beams 
of not too high energy, and special photonuclear processes may also 
play some part, particularly as secondary standards. Radioactive sources 
of gamma-rays, or gamma-ray sources which use a nuclear radiative 
transition like the decay of the familiar state of Be 8 at 17.6 Mev excita- 
tion, may be calibrated by using other particles emitted in coincidence, 
if the details of the disintegration scheme are known. The strong sources 
are usually gamma-rays produced by the stopping of electron beams by 
Coulomb scattering, or bremsstrahlung radiation. This radiation is of 
course continuous, rather than discrete, like that of the nuclear radiative 
decays. Its energy limit is in general well-known, because the original 
electron energy can be well-defined and measured. Individual quanta 
may be selected by using coincidence methods with the electron after the 
radiative act. In general, though, the whole spectrum is used, or at least 
that above some threshold for a given reaction, or above a broad limit 
set by absorbers placed in the beam to remove the softest quanta. In 
this case cross sections are hard to give for a definite energy. Again dif- 
ferential measurements may be made. Very often the cross sections for 
bremsstrahlen-induced reactions are given not per quantum of a definite 
energy, but per effective quantum in the beam. The number of effective 

quanta Q is denned (B16a) as the ratio Q = II n{E)E dE /2? maX) 

where n(E) dE is the number of quanta in the beam exposure having 
energy between E and E + dE and E max is the maximum energy in the 
beam. The rough form of the spectrum for bremsstrahlen is n(E) dE 
= N v dE/E; therefore in this approximation the value of Q for a brems- 
strahlung beam is just the number N . This is usually satisfactory if 
the region of interest is well above the low-energy deviations from the 
l/E distribution for the beam. For quantitative work the exact form 
of the bremsstrahlung spectrum must be taken into account and the 
experimental data analyzed by a numerical process indicated, for ex- 
ample, in (K4). 

Neutrons present many related problems; their detailed treatment is 
to be found in Part VII. 

5. A Table of Yields. As orientation to the size of yields to be ex- 
pected in typical practical situations, Table 1 shows a number of thick- 
target yield values for various nuclear projectiles of widely available 
energies. In all cases the target is taken to be thick enough to stop the 
entire beam, and to be the pure element of normal isotope composition. 



24 



A Survey of Nuclear Reactions 



[Pt. VI 



For gamma-rays and neutrons such a target is not usually practical, 
but computed values are given for comparison. Only a few values are 
given for reactions due to other particles than the heavy charged parti- 
cles of central interest in this chapter. Table 7 contains additional data 



Yields of 


TABLE 1 
Some Typical Nuclear Reactions (Thick-Target) 






Product Nuclei/10 6 Projec- 


Projectile 


Reaction 


tiles (beam energy in Mev) 


Gamma-rays 


AF(Y,P) 


8 X 10 s (17 Mev) 




AF(t,w) 


3 X 10 3 (17 Mev) 




Cu(7,n) 


11 X 10 3 (17 Mev) 




W( Tl n) 


18 X 10 3 (17 Mev) 


Electrons 


Cu 65 (e,e'n) 


2 (17 Mev) 


Neutrons 


n-capture on any but He 4 


10 6 




S 32 (n,p) 


~2 X 10 5 (5.8 Mev) 


Protons 


Cu(p,n)Zn 63 


18 (6.3 Mev) 


Deuterons 


Be(d,H 3 )Be 8 


230 H 3 (14 Mev) 




Li(d,2n)Be 7 


22 (19 Mev) 




B(d,n)C u 


1.5 (8 Mev) 




C(d,n)N 13 


1.4 (8 Mev) 




Na(d,p)Na 24 


190 (19 Mev) 




P(d,p)P 32 


88 (8 Mev) ; 350 (14 Mev) 




K(d,p)K 42 


11 (19 Mev) 




Ca(d,a)K 42 


0.3 (19 Mev) 




Cu(d,p)Cu 64 


330 (14 Mev) 




Y(d,2n)Zr 89 


4.9 (14 Mev); 52 (19 Mev) 




Mo(d,a)Cb 90 


0.4 (14 Mev) 




Te(d,n)I 131 


1.6(8 Mev); 32 (14 Mev) 




Bi(d,n)Po 210 


56 (14 Mev) 


Alpha-particles 


A(a,pn)K 42 


17 (44 Mev) 




Pb(a,2n)Po 210 


48 (44 Mev) 




Bi(a,2n)At 211 


0.13 (38 Mev); 130 (44 Mev) 



on deuteron reactions. Reference (GO) contains still more data on 
charged particle beams and a very useful survey of chemical methods 
for preparation of carrier-free radio isotopes. 

C. Types of Reactions : A Guide to the Literature 

It is not the function of this section to present a complete survey of the 
thousand-odd nuclear reactions which have been studied. Even a list 
of reactions by types cannot be exhaustive, since reactions have been 
observed at high energy in which dozens of products occur. It would 
be possible to systematize such highly multiple reactions only by giving 
the relative yields of the various products. Putting aside such processes 



Sec. 3A] The Nuclear Model 25 

for the present, most nuclear reactions involve the emission of two 
particles after the reaction, following the type of reaction T(i,p)Ji. A 
sizable number of reactions are known in which three or four definite 
products are emitted; then there is a continuous sequence of more 
complex reactions which typically occur at high energy. There is also 
the familiar fission reaction, in which, instead of a rather heavy nucleus 
emitting several light nuclear particles, the heavy target divides and 
two heavy fragments come off, accompanied by several neutrons and, 
infrequently, an alpha-particle. A systematic list of nuclear projectiles, 
with the products they induce, and references to original literature 
which describes reactions of the type so defined are presented in Ap- 
pendix II. The references make no attempt at completeness, nor have 
they been chosen for the purpose of giving the first papers on the reac- 
tions. They are meant to indicate a few recent and rather complete 
papers, especially those which contain theoretical treatment and good 
bibliographies. It is hoped that entries in Appendix II will assist the 
reader both in finding a rather general introduction to the reaction type 
of his interest and in getting a good start toward an exhaustive search 
of the literature. 



SECTION 3. THE NUCLEAR MODEL 

A. Qualitative Account of the Model 

We can now begin to construct a nuclear model which will have some 
connection with the complex reality of nuclear process and structure. 
In the study of nuclear reactions the comparison of the detailed implica- 
tions of successive models with experiment has led to a graphic but still 
by no means adequate model. It is this model we here seek to describe. 

It is instructive to compare our knowledge of the atom with that of the 
nucleus at its heart. We know at last that the atom is fully described 
as an assembly of rather well-defined electrons, held together by their 
electrostatic attraction to the dense central nucleus, influencing each 
other by their mutual electromagnetic interaction, and moving in accord 
with the principles of quantum mechanics (especially the exclusion 
principle). The nature and properties of electrons are well-known, all 
interactions being given in detail by Maxwell's and Lorentz' equations. 
Only for very few details — the hyperfine structure and isotope shift, 
the line shift due to electromagnetic radiation field coupling — is further 
investigation needed. Moreover, the well-marked periodicities and shell 
structure are themselves evidence that the complex mechanics of a 
many-body problem are susceptible in the atomic case to far-reaching 



26 A Survey of Nuclear Reactions [Pt. VI 

simplifications in which the many-body problem is soluble by methods 
which begin with the idea of the independent motion of a single electron. 

Contrast the nucleus. Even the forces which act between pairs of 
nucleons are still without any general expression. In Part IV of Volume 
I, the two-body problem is discussed at length in an effort to find empiri- 
cal clues to the force laws. The nuclear forces are short-range forces; 
no large-scale counterpart exists for laboratory investigation. All 
our knowledge of nuclear forces is based on nuclear experiment itself. 
The nucleus is a system of many bodies, whose general mechanics is 
evidently more complicated than that of the equivalent atom for, 
although regularities exist in nuclei, they are not so striking as those 
evident in the periodic table of the atoms. The major regularities, 
important though they are (see Part IV, Volume I), cannot yet lead to 
confidence in a simple building-up process like that of the atomic domain. 
Only in the lightest nuclei, below neon, or even below helium, is more or 
less easy progress to be hoped for in obtaining a definite and detailed 
picture of nuclear matter. We shall see how the short-range and 
exchange character of the forces, the strong component of non-central 
force, the whole tightly bound character of the nuclear state of matter 
make difficult its detailed description. The point of view so far most 
successful in the discussion of reactions, though clearly itself incomplete 
and often misleading, involves a kind of abstraction from the detailed 
structure of any particular nuclear species, and an effort to see what 
types of behavior, what experimentally determined properties, can be 
used to give an account of nuclear reactions. 

1. The Interaction of Nucleons. In the atom, each electron moves 
in a rather slowly changing force field, the sum of the nuclear attraction 
and the averaged repulsion of the other atomic electrons. The forces 
are of long-range nature, and it does not matter much if we neglect the 
very occasional close collisions between the other electrons and the one 
whose motion we are considering. We can hope to study the motion of 
each electron separately, taking account of the others present only in a 
general and smoothed-over way. The presence even of scores of electrons 
does not impossibly complicate the problem : we fix on the detailed mo- 
tion of each in turn, correcting only slightly for the specific behavior of 
its neighbors. This approximation, known as the Hartree approxima- 
tion, is excellent for the loose, smooth, open structure of the atom, gov- 
erned by long-range forces. But in nuclei the scheme seems entirely 
inappropriate. Here the forces are of a range smaller than the dimen- 
sions of the system. The force on each particle depends strongly on the 
detailed configuration of its neighbors; a small difference in the position 
of another nucleon can change its interaction from a decisive one to ;i, 



Sec. 3A] The Nuclear Model 27 

negligible one. The miniature solar system has disappeared; we think 
instead of a miniature drop of water, a small drop of quantum-mechanical 
liquid. Instead of a slowly and smoothly varying mutual interaction, 
superimposed as a perturbation upon a stronger central force, all of 
which we can without much error replace by a smoothed-over time 
average, the interaction energy of a single nuclear particle with the rest 
of the nucleus is a jagged and highly fluctuating function of time. To 
average in space over such a function is to conceal its essential features. 
The continual transfer of energy from particle to particle is the rule; 
no well-marked center of force exists. It is misleading to try to "peel 
off the motion of each particle" in turn and try to correct for the average 
effect of its neighbors. 

In this picture the shell structure of loosely coupled particles has no 
obvious counterpart in nuclei. Yet the lightest nuclei show such proper- 
ties that for a long while they have been studied by the use of the 
Hartree approximation and related methods. Moreover, the nuclei 
all the way up the table exhibit regularities, in their lowest states of 
excitation at least, which imply a well-defined shell structure. Such 
properties as (1) special stability of particular values of A and Z (of 
which the familiar increased stability of even-even nuclei is the proto- 
type), (2) the angular momentum and parity of the ground state and, 
for many nuclei, of the lowest-lying excited states (isomeric states), and 
(3) rough values of the magnetic moments of ground states can be 
rationalized and even predicted by semi-empirical results of what is 
called the shell model. 

In the most recent and remarkably successful of the shell models (M5), 
the individual nucleons are pictured as moving in Hartree-Iike shells of 
well-defined orbital angular momentum and radial quantum number. 
The order and the total angular momentum of these states can be fixed 
by arguments based on the picture of strong spin-orbit forces, which 
couple the spin and orbit of each individual nucleon strongly together, 
so that the total angular momentum and parity can be established by 
the analogue of spectroscopic j-j coupling, taken with certain semi- 
empirical rules for deciding the order of the terms. There results the 
order of shells, in the familiar spectroscopic notation, shown in Table 2. 
There is a rapidly growing literature of applications of the shell model 
to the prediction of the properties of low-lying nuclear levels; it cannot 
be doubted that no previous effort to describe such states has been so 
successful, and indeed it seems demonstrated that strong spin-orbit 
forces must be present. 

How the single-particle orbit model, with its marked shells, can be 
reconciled with our general arguments on the unsuitability of the 



28 A Survey of Nuclear Reactions [Pt. VI 

TABLE 2 

Nucleab Shell Oeder in the j-j Coupling Model 

The suffix indicating the j value has been suppressed, except when the same 
orbital angular momentum appears in two adjoining shells; then the state of 
highest j is indicated as lying lower in energy. The symbol 2p, for example, 
means both the states 2p% and 2p^. 



Shell 
Configurations 


i 

Values 


Number of 
Nucleons 
in Shell 


Total Number 

(Neutron and 

proton shells 

are independent) 


Is 

lp 

Id, 2s 

1/, 2^, lff9 /4 

lgt/i, 2d, 3s, \hm 
Xh%, 2/, 3p, liijs 


A 

A, A 

A, A, A 

A, A, A, A, A 

A, A, A, A, !K 

A, A, A, A, A, X A 


2 
6 
12 
30 
32 
44 


2 

8 

20 

50 

82 
126 



Hartree picture for the strong, short-range, and fluctuating nuclear 
forces is still by no means clear. But an interesting analogy due to 
Weisskopf seems to point out the nature of the eventual solution. He 
recalls the behavior of the Fermi gas of electrons, regarded as non-inter- 
acting, and moving in the periodic potential of a metal lattice. Here 
shells — the Brillouin zones — are very marked, and the neglect of the 
electron Coulomb interaction, certainly reasonably strong, seems to 
have very little effect. Qualitatively this must be ascribed to the circum- 
stance that electron-electron collisions can make no physical difference 
if all states in the momentum space are filled, as required by the exclusion 
principle for the Fermi gas. Such scatterings represent no change in the 
total system. If, however, a higher-energy electron enters the metal 
from without, it carries momentum adequate to excite the electrons 
of the Fermi gas beyond the Fermi limit; its collisions are rapid, and it 
loses energy very quickly to the electrons of the metal. In the same 
way, the low-lying states of the nuclear system may be represented well 
enough by the non-interacting particle picture; the expected collisions 
are mostly excluded. Yet for high excitations the regions of momentum 
space which are not filled can be reached, and the collisions become 
decisive. This picture fits the real nuclear situation very well. 

Although the stationary states of nuclei, and especially of the lighter 
nuclei, lie beyond the scope of this treatment, it is appropriate to men- 



Sec. 3A] The Nuclear Model 29 

tion here the most powerful of the methods deriving from the Hartree 
picture, which has found great use in the discussion of just those prob- 
lems. The method is the Wigner theory of nuclear supermultiplets. 
Using an extension of the same essentially group-theoretic methods 
which permit the classification of the spin and orbital quantum states 
of the atomic system, Wigner considers a nuclear system a collection of 
particles interacting by forces which depend only on spatial configura- 
tion, not on spins or on the charge of the nucleon. Exchange forces may 
occur, if they are space-exchange forces only. For such interactions, 
certain quantum numbers can be defined, in extension of the J and S 
numbers, familiar in atomic multiplets, which characterize the spatial 
symmetry of the nuclear wave functions for any collection of nucleons. 
Then, by regarding the Coulomb and the spin-dependent forces as 
perturbations capable of splitting the highly degenerate levels of the 
ideal system, a more realistic picture can be obtained which has con- 
siderable success in the ordering of the lower-lying states of not too 
complex nuclei. For specific information the extensive literature may 
be consulted. 

However, it is fundamental to the picture of the nucleus which we 
shall chiefly employ, the picture appropriate for excited states: that 
energy exchange with ease between the closely packed nucleons; that 
the nucleus be regarded as a highly condensed state of matter, a tightly 
packed quantum liquid, where the relatively small number of degrees 
of freedom (compared to those of a drop of water), and the diffrac- 
tion and exchange effects to be expected from quantum considerations 
at such small distances be thought of as modifying the behavior of a 
system otherwise very like the thermodynamic system of a familiar 
liquid drop. We know that the radius of the sphere of nuclear matter 
is rather well represented by the formula 

R = r A y ° (23) 

with A the number of nucleons, and r = 1.4 X 10~ 13 cm (CIO). This 
is just the assertion of constant density which defines our model as a 
liquid or a solid, not a gas. 

From the simplest Fermi gas model, or what is almost the same, by 
taking the mean velocity of a nucleon in the nucleus to be that corre- 
sponding to a de Broglie wavelength equal to the mean spacing of 
nucleons, we find that the kinetic energy of a nucleon ought to be some 
tens of millions of volts. This gives a mean collision time of some 10 — 13 
cm per 5 • 10 9 cm/sec, or about 10 -22 sec. For any interacting particle 
which takes a time longer than this to complete its collision with a bound 
nucleon, we have to regard the nucleus as a closed system, without a 



30 A Survey of Nuclear Reactions [Pt. VI 

sharp separation possible between collisions with a single nucleon and 
collisions with the whole of the nuclear matter. Only if collisions are 
made by particles moving so fast, and transferring so much momentum 
to the struck nucleon, that the collision is complete within the charac- 
teristic nuclear time can we think of the nucleus as a collection of free 
particles. Even then the nuclear "gas" is highly degenerate; quantum 
levels are filled up to a rather high energy, and low momentum transfers 
thus are discouraged. It is then clear that, for incident particles with 
an energy per nuclear mass unit up to 20 or 30 Mev, the nuclear drop 
must be considered as a whole. Only in the higher-energy domain are 
we more nearly justified in thinking of a collection of instantaneously 
free nucleons; and even here there are important effects of the nuclear 
binding. 

Let us continue this qualitative discussion of the collision between a 
nucleon and the struck nucleus. If the incident nucleon is of high enough 
energy so that it may be localized (i.e., represented by a wave packet 
whose main components have wavelengths smaller than nuclear dimen- 
sions), we may try to follow the collision classically. The short-range 
nuclear forces extend about as far as the mean spacing of nucleons within 
the structure. A colliding particle then has very little chance to travel 
through the nucleus without striking the nucleons within. At very high 
energies (say 100 Mev or more) the mean free path for collision with 
the nucleons of the nucleus has dropped to some 4 X 10~ 13 cm, but, 
using the same rough idea of collisions with nearly free neutrons, the 
free path has dropped to a tenth of that value for ten-million-volt 
incident particles, and to only 10~ 14 cm at 1 Mev (S7). The last dis- 
tance is so small that the picture evidently fails, but the conclusion seems 
confirmed. As soon as the particle crosses the surface of the nuclear 
sphere, it will interact strongly with the nuclear matter. It may lose 
only part of its energy to the first particles it encounters, and continue 
on, transferring energy to the nuclear matter as it goes. Eventually its 
entire kinetic energy is spread in some way over the many particles of 
the nucleus, and a new state of quantized collective motion for all the 
nuclear particles exists, a nearly stationary state of what is called the 
compound nucleus. We shall say more of this in Section 3A2. 

The nuclear drop has of course an energy content, given in an under- 
standable form by the semi-empirical formula discussed in Section 3B4. 
The electromagnetic properties of the nuclear drop may be computed 
and measured; indeed, all the properties of the ground state can be 
ascribed to the model. But the details of nuclear structure cannot be 
obtained from such simple assumptions as underlie the semi-empirical 
energy formula, especially the constant density of nuclear matter. 



Sec. 3A] The Nuclear Model 31 

Still less is to be expected from the use of this statistical model in nuclear 
reactions. In general, it may be said that what we here visualize is not 
the particular quantum level of a particular nucleus with a well-defined 
energy, but the average to be expected in a given region of mass number, 
including many similar but not identical neighboring nuclei. More 
formally, we shall deal with all the characteristic properties of nuclear 
states not one by one, but averaged over an interval of excitation energy. 
Only if the fluctuations in these properties are not too great from level 
to level, so that the average behavior represents the individual behavior 
of states, or if the experimental situation produces averages as the quan- 
tities directly observed, can we hope for satisfaction from such a statisti- 
cal point of view. It is in this charitable sense that the results of the 
theory are to be taken. 

We can summarize the above statements by setting out the two leading 
assumptions under which the results of the present theory of nuclear 
reactions are obtained: 

1. Nuclear matter is composed of closely packed particles, strongly 
interacting and capable of rapid interchange of energy, like the particles 
of a drop of some quantum-mechanical liquid. We may call this 
assumption close packing. 

II. The results obtained from such a consideration are meant to 
apply not to the particular and specific properties of one well-defined 
state of motion, but to an average over many states of neighboring 
energies. Only if the property studied does not fluctuate too widely 
in such an interval will the conclusions be trustworthy. This we may 
call the statistical assumption. 

2. The Compound Nucleus. In Section 3A1 we discussed the dissi- 
pation of the kinetic energy of some incoming particle as it passed into 
and merged with the closely packed particles of the nucleus. The 
resulting configuration we called the "compound nucleus." Let us go 
on with the story of such an event. After the incident particle has 
merged completely with the nucleus, its energy, no longer concentrated 
in one particle, is shared by the collective motion of all particles of the 
new system: the compound nucleus formed by projectile and target. 
Each of the nucleons will have some additional energy, but none will be 
likely to have all the incident energy at any one time, or even any very 
large fraction of it. The energy will be spread among the very many 
degrees of freedom exactly as the heat energy of a drop of water is kinetic 
energy spread among many degrees of freedom. But the compound 
nucleus will eventually lose its energy of excitation. Either radiation, 
which, as we shall see, is a relatively slow process, will finally remove the 
energy and thus "cool" the nucleus, or after a long time (time enough 



32 A Survey of Nuclear Reactions [Pt. VI 

for many collisions of nucleons within the nucleus, some ten million in 
typical cases) the configuration will have the rather unusual property 
that a sizable amount of energy is concentrated on a single particle, and 
a particle may escape, cooling the nucleus by "evaporation." It is 
evident that the escaping particle will not in general be the particle that 
entered, and that it will have neither the energy nor the direction of the 
incident particle. Even if by chance a neutron, say, goes out when a 
neutron enters, this need not be an elastic collision. On the contrary, 
excitation energy will in general be left behind, to be dissipated in another 
way; the collision will be inelastic. Elastic collisions do not take place 
with formation of a compound nucleus except by the re-emission de- 
scribed under the special and rare circumstance that the residual nucleus 
is left in the single ground state only. The elastic scattering observed 
includes other effects by which the incident wave function is scattered 
coherently without the actual formation of a compound nucleus, through 
purely wave-mechanical processes. 

The formation of a compound nucleus may, it is true, take place 
under rather special circumstances. A gamma-quantum, with wave- 
length long compared to the nuclear size, may excite proton vibrations 
with a definite phase relationship. Or, considering the same process 
from a quite different point of view, the quantum may act upon a single 
surface proton which receives enough energy to escape without striking 
other nucleons on its way out. Or the proton coming out may lose 
energy to a single spot on the nuclear surface, "heating" the nucleus 
not as a whole but locally. The final reaction might occur by evaporation 
before the "conduction" of heat away from the hot spot to the rest of 
the nucleus has taken place. In all these cases — we believe them infre- 
quent but possible — a detailed consideration of the particular process 
(B17, Kl) would be required. However, we shall discuss mainly the 
fundamental notion of Bohr, which is that the compound nucleus is 
capable of disintegration in a manner independent of its method of 
formation. This is perhaps the third assumption of our theory, and we 
have seen under what circumstances it would require modification. We 
shall state it explicitly: 

III. Nuclear reactions in the domain of statistical theory take place 
in two separable steps: (1) the formation of a compound nucleus by 
combination of the incident particle and the target nucleus, and (2) the 
subsequent disintegration of the compound nucleus, in a manner inde- 
pendent of its method of formation, into the products of the reaction, 
the emitted particle and the residual nucleus in some one of its quantized 
energy states. A reaction that leads to more than two products proceeds 
by a continuation of this scheme: the first product comes off, and the 



Sec. 3A] The Nuclear Model 33 

residual nucleus then acts as an excited compound nucleus, emitting 
another particle, and subsequently even another, and so on. Thus 
as many as a dozen neutrons may be "boiled off" successively from a 
highly excited nucleus (T4). 

3. Nuclear Energy Levels and Level Widths. So far we have dis- 
cussed the ideas of the statistical theory from the point of view of the 
close-packed nucleons, an almost geometrical and quite classical argu- 
ment. The same conclusions can be expressed in terms of energy levels 
rather than in geometric terms. We have already shown that the com- 
pound nucleus exists for a time long compared to the mean free time for 
a nucleon collision in the nucleus. This means that, although the descrip- 
tion of the individual nucleon as free with a definite energy was very 
rough, the compound state can be thought of as existing with a wave 
function which is the approximate eigenfunction of the Hamiltonian 
of all the nucleons. We restrict ourselves to states which genuinely 
represent the compound nucleus or its disintegration products: that is, 
to wave functions which contain no parts corresponding to one particle 
at a distance from all the others, except where the distant particle is 
represented by an outgoing spherical wave. This serves to exclude 
states in which a free particle exists which could not have been emitted 
by the compound state. Under these conditions we can write 

*. = ^ n -e [E "^ ir " /2)m (24) 

where the energy is no longer a real eigenvalue but contains an imaginary 
part. This is the familiar procedure for a damped oscillator (H6). 
The fraction of systems in the state n will vary with time like the factor 
e -IW \ Thus T n /h is the reciprocal mean lifetime of the state; V n is 
called its width. The usual expression of the uncertainty principle 
AE • At c^ If is exemplified here. 

What does the spectrum of a nucleus look like? Simply by considering 
the distribution of energy among the many particles of a tightly bound 
system we can form some idea. We begin with the fact that the binding 
energy of a particle in the nucleus is only a few times smaller than the 
rough estimate we can make of its kinetic energy. If the particles did 
not interact strongly — a bad assumption — we could think of forming 
the excited levels of the nucleus by giving all the excitation energy to 
one nucleon, or dividing it among two, three, • • • , and so on. Now, 
since the well in which we could imagine the particles to be moving (see 
Fig. 3) is far from fully occupied with particles, we can find levels for 
each nucleon which lie quite close to its ground level. Thus the dis- 
tribution of a few Mev of excitation energy could be accomplished in a 
very great number of different ways. Most of these would correspond 



34 



A Survey of Nuclear Reactions 



[Pt. VI 



to the general spreading out of the excitation energy among the many 
nucleons; very few indeed would be the states in which all the energy 
was used to promote one or even a few nucleons. The coupling of the 
particles of course means that the levels are of mixed character. A level 
cannot be ascribed to a specific distribution of energy, spread among 
a definite set of particles. In any state, part of the energy is sometimes 
concentrated on one, sometimes on two, etc. Each eigenfunction could 
be expanded into a series of terms corresponding to the various partitions 
of energy. But, with so many possibilities available, the contribution 



E 




Continuum 
(dissociation) 



rz — -^ 




:l=5S?s=: 




_ ~ ~ 


Empty levels 








> Filled levels 
















■ 


y 



Effective 

potential 

well 



Fig. 3. Schematic nuclear potential well and nucleon energy levels. 

of a partition which corresponds to assigning all the energy to one or 
even two nucleons will be very small indeed. The total number of levels 
will evidently be extremely large. 

These levels, as we have said, are not completely sharp. Only the 
ground level is quite sharp (even it is widened imperceptibly by the 
possibility of beta- or alpha-decay or of spontaneous fission). As we 
add excitation energy, the levels become wider because of possible 
radiative transitions to the lower states through emission of quanta. 
For some sufficiently high excitation energy, the nucleus will be capable 
of emitting one particular particle, say a neutron. This will, of course, 
represent a widening of the levels. But, since so little of the wave func- 
tion represents a configuration with all the available energy concentrated 
in one neutron, the effect will be by no means large. The level width 
will not change markedly as a new decay process appears. The levels 
will simply widen progressively as more and more possibilities of decay 
become energetically allowable. The character of the levels will not 
change abruptly with their energy. Only the appearance of a small 
probability of emission of a particular particle will indicate the crossing 
of the energy threshold. As the energy of excitation increases, the total 



Sec. 3A] 



The Nuclear Model 



35 



width of levels rises. Moreover, the increase with energy of the number 
of ways the energy can be partitioned is marked. Finally, the spacing 
is so small and the width so large that the levels overlap and the familiar 
continuum has been reached. Even here fluctuations in level density 
and special selection rules may maintain the features of a discrete 
spectrum. Only when the excitation is comparable to the total binding 
energy — say 6 to 8 Mev per particle — does the character of the levels 
change qualitatively. In such a region the concentration of sufficient 



-t 



>- Continuum 
(dissociation) 



A few empty levels 




Atomic 

potential 

well 



Fig. 4. Schematic atomic potential well and electron energy levels. 

energy in one nucleon to allow its escape is the rule and not the exception. 
Under these circumstances a genuine continuum of levels exists. 

In Fig. 4 is shown the parallel situation in the case of an atom with 
many particles. From the geometrical point of view, the atom is a loose 
open structure. Its long-range forces mean that elastic collisions are 
the rule; even if an incident particle excites the atom, the energy of 
excitation is given to one or perhaps a few of the valence electrons. The 
K shell electrons have in a heavy atom about a hundred thousand volts 
of binding; the valence electrons, only a few. Even the L electrons are 
bound only by some tens of kilovolts. The possible energy levels thus 
are as shown in the figure: very narrow bands of close levels with great 
empty spaces separating the bands. Thus in general there is little 
chance that the energy of an incoming electron lies in a region where 
levels of the compound system exist. For atomic collisions these nearly 
stationary states are of little importance. Almost any system which 
has energy enough to release an electron will rapidly do so. It is inter- 
esting, however, that, even in a system so simple as the beryllium atom, 
terms have been found which give rise to broadened lines. These terms 
correspond to states in which the total excitation of the atom is greater 



36 A Survey of Nuclear Reactions [Pt. VI 

than the ionization potential but is divided between two electrons. 
These states often have large widths and short lifetimes, because they 
can decay by radiationless transitions, emitting electrons. The analogy 
to the nuclear case is close. In molecules, of course, predissociation is a 
prominent effect. 

Although the concepts of level spacing and level width are entirely 
quantum-mechanical, Weisskopf has suggested a very pretty semi- 
classical picture of their significance. Consider a complicated many- 
particle system like a nucleus, but for simplicity with a large number of 
equally spaced levels, with energy of the nth level, E n = E + nS. We 
can make up a wave packet by combining a great number of energy 
eigenf unctions <p„(r) corresponding to the stationary states E n . The 
total wave function is then 



N / N \ 

; V „ -iEn*/t I V" 1 „ -inSt/h\ 

t = 2-, a-nVne =1 L a n <p n e J 



-iBof/k 



Now it is evident that 

/ 2xh\ 2 . 



( 2xh' 



Thus we have constructed a wave function which is by no means a sta- 
tionary state; in it the configuration of the system repeats itself after 
the lapse of a time t, which we can call the recurrence time. If the level 
spacing is 5, the recurrence time is given by the relation t = 2irh./8. Of 
course, a real nucleus has levels which are not equally spaced, but a sim- 
ilar physical interpretation may be given to the quantity D, the mean 
level spacing at a given excitation energy. Closely spaced levels imply 
long times of recurrence, and conversely. It is also instructive to con- 
sider the distance traveled by an identifiable nucleon of the wave packet 
during the duration of one period of the motion. If the mean velocity 
of the nucleon is given by something like a Fermi gas model, chiefly 
the zero-point motion arising from the confinement within the nuclear 
volume, we get for the velocity v = h/MpF. Then the path length I 
traveled by a nucleon which had, say, just penetrated the nuclear surface 
from without before it found itself back in the original configuration 
again would be I = 2irh 2 /MDp F . Taking a level spacing of 10 ev, 
appropriate for the region of slow-neutron capture in heavy nuclei, the 
value of I ~ 10 — 6 cm, which is about a million times the nuclear diam- 
eter. This makes picturesque the complex motion in a compound state, 
and adds some confidence to our ideas about the statistical theory. 
On the other hand, if D = 1 Mev for the low-lying states of light nuclei, 



Sec. 3A] The Nuclear Model 37 

the recurrence path is only a dozen diameters or so, and the picture of 
an orbital motion for a single particle becomes plausible once more. 

The level width can be given a similar interpretation. If a state has 
energy enough to emit a nucleon, we can imagine the packet so built 
as to describe the configuration existing when a given nucleon has just 
entered the nuclear surface. Now, after the recurrence time t, the 
nucleon would be back at the surface again, with the same energy, ready 
to leave in just the manner in which it entered. If it were actually to 
leave at this first recurrence, the implication would be that the lifetime 
of the state, h/r, was just equal to the recurrence time. Then the spac- 
ing and width would be of the same order, and the idea of well-defined 
compound state hard to justify. This is plainly one limiting case. But 
we must recall that at the nuclear surface there is a sharp change of 
potential. Within the surface, the strong nuclear forces act; outside 
(a distance only a couple of times 10 -13 cm outside), there are no nuclear 
forces at all. At this potential jump, the particle de Broglie wave suffers 
reflection. Suppose that the probability for penetration of the barrier 
is P. Then the width can be given in terms of the recurrence time: 
T ~ h/(r/P); and hence, in terms of the level spacing, T ~ P-D/2t. 

For the case in which the barrier is entirely nuclear (no Coulomb or 
centrifugal forces, realized by an s wave neutron), the penetrability is 
simply given by the familiar quantum formula P = 4k/K, where the 
quantities K and k are, respectively, the wave numbers for the nucleon 
de Broglie wave within and without the nuclear surface, and we have 
gone to the limit k « K. This is a very important result of our detailed 
theory, to be discussed later at length. Of course, in the present account 
we have disregarded the fact that there are alternative modes of decay, 
and the fact that there are constants of the motion besides energy, like 
angular momentum and parity, which must be conserved but which may 
vary from level to level. In the calculation of the appropriate D, such 
classes of states are to be regarded as contributing independently. In 
general, the statistical theory we discuss here is limited to the treatment 
of levels en bloc, without much hope of identifying states of a specific 
kind or of predicting energy levels. Since it is clear that the many well- 
defined observed levels in thermal neutron absorption in the heavy ele- 
ments (spaced as they empirically are some 10 ev, and lying 8 Mev above 
ground) must be not far from millionth members of some series of terms, 
it is not unexpected that we would make more progress, in heavy nuclei 
at least, by some general statistical treatment than by a method based 
on classifying and identifying individual terms. 

Individual levels can be observed at low energies in all nuclei, e.g., 
wherever gamma-ray spectra can be studied. In the lighter nuclei, such 



38 



A Survey of Nuclear Reactions 



[Pt. VI 



levels are visible up to tens of millions of volts of excitation; in heavier 
nuclei, not more than a million volts beyond the neutron binding energy. 
The most general features of observed level distribution is the steady 
increase of level density with excitation energy by a function which 
becomes the steeper as we go to heavier nuclei. A few examples of 



10' 



_ / V N 15-11.7 



hJ3 



a) 
ft 



> 

CD 



10 ; 



Observed averages for 
zero excitation 



A 

F 20-6.7 
Na 24-7.2 
~ Al 28-7.9 




8Mev 
Observed points: 
Z-A -excitation energy 
Curve models: 

— - ■■ ■ Semi-empirical 

- — — «— Fermi gas 
_ - — Combinatorial 



o 



100 



200 



Fig. 5. Nuclear level spacing. This figure presents the nuclear level spacing as a 
function of the mass number A for various nuclear excitation energies. The marked 
curves are plots of the results of various models discussed in the text. The circles 
mark individual observed levels; the nucleus and its excitation are marked beside 
each experimental point. The three horizontal lines show the spacing observed at 
zero excitation, averaged over a fair number of nuclei in each region of A. The 
roughly satisfactory nature of the theoretical curves is shown, as well as their inade- 
quacy for both light nuclei and heavy nuclei, like Ni and Pb in which the influence 
of shell structure is marked, especially at modest excitation energy. 

estimated average spacing, taken from the few known levels in the 
various regions of interest, are given in Fig. 5. It is noteworthy that the 
average spacing near zero excitation decreases from the lightest to the 
heaviest nuclei by a small factor, perhaps by about a factor of 10, while 
the spacing at some 8 Mev excitation changes by five orders of magni- 
tude. It will, of course, not escape the reader that the application of the 
statistical notion of level density to the few levels of the lightest nuclei 
is a dubious business. 



Sec. 3B] The Nuclear Model 39 

The width of the levels increases as their spacing decreases. We 
expect levels that overlap one another in particle excitation of reasonably 
heavy nuclei some 10 Mev above the ground state. Unfortunately the 
difficulty of producing closely monoenergetic beams with high kinetic 
energy has limited the possibilities of experimental check. The use of 
endothermic neutron-yielding reactions has been about the only direct 
test of this point. The work of Barschall with total neutron cross sec- 
tions at energies around \ Mev on many elements indeed serves notice 
that this picture is too simple (B3). The levels vary greatly in spacing 
and strength ; for certain nuclei the whole level structure is far too coarse- 
grained for the validity of our picture. This is primarily a result of the 
fact that the shell structure strongly modifies the binding energy of the 
added neutron. Excitation by neutron capture with low kinetic energy 
may excite some nuclei (those with closed shells) by only a fraction of the 
usual 8 Mev. At the lower excitations, the level density is much reduced. 
Other reasons may also be involved. 

B. Calculation of Level Densities 

Again disregarding certain details of structure, we may proceed to 
calculate mean level densities for some statistical models of the nucleus. 
We shall present two of the most instructive in some detail, giving only 
a sketchy account of some other schemes which have been proposed. 

1. Combinatorial Model. We have already discussed qualitatively 
the simple model for which we here calculate the mean level density. 
It is based upon the idea of a set of independent particles among which 
we divide the energy of excitation. Suppose that each of the N particles 
has an identical energy spectrum, which we shall take to be of the sim- 
plest possible form: a set of equidistant levels of uniform spacing 8. 
Then the excitation energy W can be written as 

W = £ m& 



i 



where the n t are integers showing which level of its own spectrum each 
particle occupies. A given energy W can then be realized in as many 
different ways as there are ways to sum integers to a given integer total. 
The states thus formed would be highly degenerate, the degree of de- 
generacy being equal to the number of ways in which the fixed sum can 
be realized. If we now take into account the interaction of the particles, 
the degeneracy will be split, and each level will be split into as many 
levels of the whole system as the value of its degree of degeneracy. In 
this way the level density can be calculated. The result is based upon 
the asymptotic formula of Hardy and Ramanujan (H4) for the number 



40 A Survey of Nuclear Reactions [Pt. VI 

of different ways, p(n), to form a given number n by any of the possible 
sums of smaller integers: 

p( n )=_J_ „*<*»>* 

4(3) ^n 

If we take as a value for the unit level spacing 5 the average observed 
separation between the lowest levels of heavy nuclei, 2 • 10 5 ev, we get, 
for the degree of degeneracy expected for a total excitation energy W 
of 8 Mev, 



Q- 



pi—) = p(40) = 4-10 4 

which gives an average level separation for the whole excited nucleus 
of about 5 ev, which very roughly corresponds to the density of level 
separation estimated from slow-neutron collisions. The level separation, 

4.(3)^We~' iH ' w/s)>i 



p(W/sy 



depends exponentially on the excitation energy; this is characteristic 
of all nuclear models and is displayed even by this very simple one. 

2. Gas-Like Models. In this model the nucleus is visualized, in the 
first approximation, as consisting of gas — a set of non-interacting par- 
ticles — held in a spherical box of nuclear size. The two Fermi gases 
present (neutrons and protons) occupy all the lowest levels in the box. 
As the nucleus is excited — "heated" — some of the higher levels are 
occupied with increasing excitation. Bethe has given a careful statistical 
calculation for this model, following the usual methods of statistical 
mechanics (B13). He has used a statistical ensemble with energy, 
particle number, and angular momentum taken as given constants. 
The calculation is of course quite parallel to the calculation of the entropy 
of the system, since entropy is proportional to the logarithm of the 
number of states of the system. Somewhat greater care is required for 
the computation of the number of states than for its logarithm. If W 
is the energy of excitation and A the number of nucleons, he obtains, 
for the mean spacing D (in ev) of nuclear levels of some definite angular 
momentum /, 

D = 5 .ltf . (^}\ e-Uw/2.1)" (25) 

In this formula we have assumed the usual r = 1.5 • 10" 
is in Mev. 



Sec. 3B] The Nuclear Model 41 

A similar but more sophisticated model was proposed by Bardeen (B2). 
He solved essentially the same problem, but he made use of the proper- 
ties of the exchange forces in nuclei. It has been shown that the two- 
body problem in a square well with exchange forces could be reduced 
to the one-body problem except that the potential acting on one particle 
now depends on the wave number of the interacting particle. It turns 
out that the total energy of the equivalent particle motion increases 
more strongly with the wave number than the kinetic energy alone. 
This effectively increases the phase volume available for levels, reduces 
the number of individual particle states, and therefore much more 
strongly reduces the density of levels of the whole system. The effect 
is quite analogous to the effect of the correlations of electrons which 
obey the exclusion principle in changing the interaction energy of elec- 
trons in a metallic lattice. The model of Bardeen is often referred to as 
the free-particle model with correlations. The energy dependence of 
the result is given by Eq. (25) with (ATT/2. 1) replaced by (ATT/4). 

3. The Liquid Drop. The idea of regarding the nucleus as a classical 
drop, with volume energy and surface tension, is by now a familiar one. 
On this basis the normal modes of vibration of a liquid drop may be 
computed: there are surface vibrations and transverse and longitudinal 
volume waves. The surface energy and the compressibility of the 
nuclear matter may be estimated from the semi-empirical energy content 
given in Section 10. Then the vibrations of each normal mode are 
thought of as quantized, and energy distributed among them according 
to Planck's distribution law. The nuclear matter is so "cool" that 
equipartition is far from achieved; the excitation energy is not very large 
compared to the zero-point energy. From the treatment of Bethe, 
numerical results can be obtained (B13). Here again the energy de- 
pendence is given roughly by the formula 

Z)~exp (-cTT M ), TT small D ~ exp (-c'TT M ), TT large (26) 

It is interesting that in this treatment one can show that only a few 
normal modes of surface vibration are present for heavy nuclei in the 
region of 1-Mev excitation or lower. The possibility of rotation of the 
drop, associated of course with its angular momentum and the multipole 
moments of its low-lying states, has been considered. Though the idea 
is a satisfying one, it can be said that the drop model does not account 
quantitatively for the low-lying levels of high angular momentum which 
seem to be responsible for nuclear isomerism. On the contrary, direct 
calculation of the spacing of rotational levels gives for their energy, for 
a lead nucleus say, only a few kilovolts. Such fine structure has not yet 



42 A Survey of Nuclear Reactions [Pt. VI 

been observed. It has been pointed out that particles whose wave func- 
tions must satisfy symmetry properties may, unless very rigidly bound, 
be unable to rotate as a rigid body rotates because some rotational 
states of motion are not allowed. Indeed, it has been shown (T3) that 
only for high angular momenta, of the order of 50 units, will nuclear 
rotational levels have the order normal for a rigid body or a classical 
drop rotating as a whole. At such angular momenta, the excitation 
energy is some Mev, and vibrational motion is of course important. 
The suggestion that a drop or a solid be used to represent the lowest 
states of nuclei has not proved very satisfactory; as a matter of fact, the 
magnetic moments of ground states are much better represented by the 
idea of an odd particle rotating about a non-rotating core containing 
all but the odd nucleon (S4). Again, we may expect better results for 
the drop model in highly excited states than near ground. A formulation 
of the structure of nuclei, where individual level properties are decisive, 
is still beyond the scope of the present models, although for the lowest 
states the individual-particle shell model in j-j coupling form has had 
remarkable success. 

4. Thermodynamic and Semi-Empirical Results. The general behav- 
ior of the level density as a function of energy is similar in all the models 
discussed. The details are not correct, and the constants deduced com- 
pare indifferently with experiment. It seems reasonable then to compute 
a general form of the level density as a function of energy from the most 
general statistical or "thermodynamical" point of view, and to adjust 
the constants until we get reasonable agreement with the not very well- 
known level spacings observed. 

Let us begin with the fundamental statistical-mechanical relation for 
the sum over states (B13) 



z=Z. 



-Ei/r 



where the values E t are the energies of the states of the whole system, 
and r is the parameter which we can identify with the usual kT, the 
temperature measured in units of energy. Since we are dealing with 
dense levels, we can rewrite the sum as an integral : 

Z=Jp(E)e~ }s ^dE (27) 

But let us introduce the function A, the Helmholtz free energy of thermo- 
dynamics, by the relation: Z = e~ AlT . Now, if the system has a large 
number of degrees of freedom and the distribution is one of equilibrium, 
we expect that the contributions to the integral in (27) come mainly 



Sec. 3B] The Nuclear Model 43 

from the region near the maximum, which we call W, the mean excita- 
tion energy of the nuclear system. We then rewrite the expression as 

Z = Ce ln '>- ElT dE 

/a(lnp) a'lnp (ig-WO 2 i -(g-WQ 
e -al- (E_,,0+ -8^ 2 + ; rf# 

Expanding the integrand about its maximum by the familiar saddle- 
point method, we obtain 

w , T /a 2 (ln P -£?/r)| 1* 

Z = P (W)e-^/' • -2x/ * \ = e (28) 

L / ah \w-i 

Now let us introduce the familiar thermodynamic function, the entropy 
S, by the relation 

A = W - tS (29a) 

We expect this function to be proportional to the logarithm of the den- 
sity of states. From (29a) we get the elementary definition of entropy : 

dS 1 

= - (29b) 

dW r 



and we can use this in (28) with d(lnp)/dE \ W = 1/t, to get 

e S(W) 

o(W) = 

T{2wdW/drY A 

We then have, for the general connection between density of states 
p = 1/D and the thermal properties of the system, 



J." 



dW'/r 

P (W) = v (30) 

K t(2tt dW/dr) 1A 

To determine the level density for any system, then, we need to find the 
entropy of the system, which we can always do by integrating (29b), 
if we know only the expression for the heat content as a function of 
temperature, that is, the relation between excitation energy and the 
temperature parameter t(W). 

Next, the procedure is either to adopt a model, and so work out the 
heat content as we have indicated in the previous sections, or to make 
some general assumptions for the temperature dependence of the heat 
content, and then fit the constants to the data. 

We can proceed in a general way by using thermodynamic arguments. 
We expect the heat capacity for nearly any system to be a steadily 



44 A Survey of Nuclear Reactions [Pt. VI 

increasing function of the temperature. If the temperature of a system 
is so high that for all the degrees of freedom the energy r is large com- 
pared to the quantum energy ho> , the classical limit of equipartition is 
reached, and the heat capacity is constant. But the nucleus, as we have 
said before, is generally rather "cold." The simple model of the Fermi 
gas of A nucleons in volume -f ttR 3 yields the result (M4) that the "gas" 
is degenerate if the excitation energy is small compared to Ah 2 /2MR 2 
~ 8A H Mev. Then heavy nuclei are highly degenerate below excitation 
energies of some 50 Mev. The interaction of nucleons does not affect 
this conclusion, just as the electron gas in metals is degenerate in spite 
of the electron-electron forces. 

Now we can write the heat content as a function of temperature in 
the power series: 

W = c + c lT + c 2 t 2 -j 

From the general quantum ideas of the third law of thermodynamics, 
we expect the specific heat to vanish as T — > 0. Thus the expansion of 
W near zero temperature, i.e., in the strongly degenerate region, ought 
to begin with at least a quadratic term, with all higher powers negligible. 
The familiar Debye specific heat for metals x actually vanishes at low 
temperatures with an even higher power of the temperature, since the 
heat content goes to zero like 2 14 . If we assume then any simple power 
law dependence, W = cr n , we can integrate 

dS _dS dr 1 

dW~ drdW~ t 
letting 

n 



S = 



(n-1) 
With this expression, we obtain for p(W) the value 

> m " ^ (r^)"' 2 «"■' (<dhj ' c "" w """) 

For the Fermi gas, the exponent contains 1 — K, for the liquid drop 
the power W* to W H at low and high temperatures respectively. 
It seems appropriate to use the very simple form 

P = Cexp(aJF)^ (31 a ) 

If we do this, and try to make a rough fit to the data, we obtain (following 
Weisskopf) (W5) for nuclei heavier than A = 60 a result which is approx- 
1 See (M4), p. 253, for formula; p. 248 for derivation. 



Sec. 4] Nuclear Level Widths 45 

imated by the simple form, for nuclei of odd or even Z and N — A — Z, 

1 12 

~ ^odd-odd ~ (-"even-odd ~ ^ odd-even = *b even-even = T~. T (JVIev) 

2 (.4-40) 

a = 3.4(A - 40)^(Mev)- 1 (31b) 

We present in Fig. 5 a few selected data on observed level spacing, 
together with the values calculated from the three nuclear models 
described. There is some inconsistency in what the figures represent, 
for the experimental data from slow-neutron capture, as well as the 
free-particle model calculation, refer to levels of a definite angular mo- 
mentum (or within ±Jh from the ground state). The thermodynamic 
calculation certainly includes all levels. Here there is a chance to add 
some more definite assertion to the general statistical notions; all we 
can do at present is regard all levels as more or less uniformly distributed 
in regard to most not very large values of angular momenta. This 
vague statement is typical of our statistical theory, and it may account 
for discrepancies observed in individual cases where specific angular 
momentum properties may be involved. The least we can do to recog- 
nize the great importance of nuclear energy shells is to use differing 
constants for odd- or even-Z nuclei. 

SECTION 4. NUCLEAR LEVEL WIDTHS 

We have already mentioned the concept of level width, which is defined 
as r„ in the expression for the approximately time-independent (because 
long-lived) quasi-stationary state wave function for the compound 
nucleus, 

From this form for the decaying amplitude of the wave function the 
width can be related to the mean lifetime in the usual way: r n = h/i-„. 
This expression naturally refers to the change in amplitude with time 
of the compound state regardless of the products of its decay; it is the 
total width, and r n is the total lifetime. Naturally, the compound state 
may in general decay in more than one way; a single set of reactants 
may have many alternative sets of products. We may generalize the 
notion of width to include the idea of partial width by writing T n 
= 2 rVa(w), where the r ta are the partial widths, which we index here, 

ia 

somewhat redundantly, with two indices: i, which designates the kind 
of particle emitted, as neutron, gamma, proton, etc.; and the index a, 
which gives the state of the residual nucleus. It is evident that the same 



46 A Survey of Nuclear Reactions [Pt. VI 

particle can be emitted with different properties, as different energy or 
angular momentum, leaving behind generally, though not necessarily, 
a different state of the residual nucleus. The argument of the function 
Ti a (ri) refers to the state n of the compound nucleus whose width is here 
written. Clearly the partial width for the emission of, say, a neutron 
to any possible state of the residual nucleus might be the experimentally 
useful quantity, and it would involve the sum ^ r neutpa (n) over all 

a.neut 

accessible states which can be left behind after a neutron is emitted. 

To speak somewhat academically, r„ is strictly zero for almost no 
states of nuclei. Even the ground state is not wholly stable; in radio- 
active nuclei it may decay by beta- or alpha-emission, and in nuclei not 
observed to be radioactive we know that in general a state of less energy 
content can be reached either by a process like spontaneous fission or, 
if the nucleus is a rather light one, by combination with a neighboring 
nucleus of the matter present. Such lifetimes are long, perhaps enor- 
mously long, compared to actual nuclear reaction times, under terrestrial 
conditions at least. It -is conventional therefore to disregard such forms 
of radioactive decay, and especially beta-decay, and to regard the levels 
as widened only by the possibility of radiation or by the emission of a 
heavy particle. This implies that, while I^ may be different from zero, 
its study belongs properly to radioactivity theory, and not to the dynam- 
ics of nuclear reactions. The levels of a typical nucleus then can decay 
only by gamma-emission, which slowly widens the states as their excita- 
tion increases until the excitation energy reaches a value which makes 
heavy-particle emission energetically possible. For nuclei whose ground 
states are empirically alpha-radioactive, of course, this condition is ful- 
filled for every level, but for most nuclei from the middle to the top of the 
periodic table such a value is reached only at the binding energy of a 
proton or neutron. The binding energy lies near 8 Mev for the entire 
middle of the table; for the lightest nuclei, up to neon, say, it is widely 
varying; for the nuclei from gold or lead and up, it has fallen to 5 or 
6 Mev (W3). Up to a level of the compound nucleus, then, which has 
an energy content at least 5 to 8 Mev above the ground state, only 
gamma-radiation will introduce a finite width; beyond it, first the 
emission of one particle, then of another, and finally of even more kinds 
and a greater number of particles will widen each level. Moreover, not 
only do the values of T(n) increase because there are more possible I\-, 
but in addition more states are available in the residual nucleus, so that a 
takes on more possible values. The widths themselves increase slowly, 
as we shall see, with the kinetic energy of the emitted particles and the 
energy of the gamma-rays. 



Sec. 4A] Nuclear Level Widths 47 

A. Level Widths and Reaction Cross Sections : Statistical Relations 

We have already indicated that the fundamental idea underlying the 
present statistical treatment of nuclear reactions is the separation of the 
process into two independent steps: the formation of the compound 
nucleus, and its subsequent independent decay. Upon the basis of this 
assumption there can be derived a useful relation between level widths 
and cross sections. The method of the derivation is the familiar statis- 
tical-mechanical one of detailed balancing (W5). 

Let us consider an equilibrium mixture of three kinds of systems: the 
compound nucleus we are studying, and its two decay products, the 
"emitted particle" and the residual nucleus. The equilibrium reactions 
are the decay of the compound, and its formation by recombination of 
the two decay products. We may write 

C* *± p + R a 

We do not consider every possible state of the residual nucleus R and 
the particle p which might combine to form the given state of the com- 
pound nucleus, but only particles of such energy, angular momentum, 
etc., that they can combine with a residual nucleus ~R a in a definite state 
a to produce a particular state of the compound nucleus C*. It is clear 
that we must be sure of a random statistical ensemble: we must be cer- 
tain that special phase relations do not exist between the wave functions 
of the decay products and the compound nucleus which emits them or is 
formed from their combination. To insure this, and to free ourselves 
from the possible special character of individual states of the compound 
nucleus, we have to regard the levels of the entire system as very closely 
spaced, and be sure that the energy interval is large enough to include 
many levels of the system. Our widths and cross sections are then aver- 
ages over this small interval of energy, small compared to the scale of 
variation we- are interested in, but large enough to include many levels. 
All phase relations are then lost in the averaging. This is .the usual 
procedure required to define a quantum-statistical ensemble. 

Now we apply the almost revered formula of the time-dependent 
perturbation theory, which gives the transition probability per unit 
time for the transition from the almost stationary state (A) of a system 
to a continuum of states B at the same energy. This is 

2tt . ., 

w A -,B •= —\Mab\pb® 
h 

where [ Mab \ 2 is the square of the matrix element of the perturbing 
operator between the initial and final states involved (the final state is 



48 A Survey of Nuclear Reactions [Pt. VI 

chosen as any one of the continuum having the correct energy), and pb 
is the density of states of the final system, per unit energy in the volume 
within which the wave functions are normalized. Now the inverse 
process is given by the same formula written for w B ^ a- But, since 
the operator is certainly Hermitean, | Mab \ 2 = \ M B a | 2 - We then 
obtain the expression for detailed balancing in its general form 

WA ^B-PA = W B ~,A'PB (32) 

The use of perturbation theory conceals the generality of the result; any 
order of the perturbation will give the same dependence on the absolute 
square of a Hermitian matrix element, so that the result is quite general. 
Applying it to the situation described above, with a compound nucleus 
of excitation energy W decaying to two reaction products, we may write 

Tpa(W) 4TP 2 dp 

— p c (W) = Vffpa-ga. — — • — - 

n (2irhy dW 

Here we have used the relation between width and lifetime, w = 1/V 
= r/h, and found p B by using the density of states for a free par- 
ticle with mass m p , momentum p, velocity v, and kinetic energy E p , 
47rp 2 /(2irh) 3 , and converting the density in momentum space to the 
density in energy by the factor dp/dW. Note also the important relation 
w = v<x for any collision between two particles of relative velocity v, 
in normalizing volume. The factor gr R is the statistical weight (degree 
of degeneracy) of the residual nuclear state. We may finally write 

T pa (W ) PC (W) =^-~ with X = — (33) 

2jt 7rX m p v 

This formula tells us that the mean level width, measured in units of the 
level spacing, is just equal to the cross section for the inverse reaction, 
a pa , measured in units of the maximum cross section for absorption of 
particles of the given kinetic energy, apart from small statistical weight 
factors. It is interesting to recall that the maximum absorption cross 
section for s waves is just 4irX 2 , and thus the width for s wave emission 
is at most 2g R /w times the level spacing. 

The use of the detailed balancing formula for connecting lifetimes to 
cross sections as demonstrated here is only one of its applications. By 
considering an over-all reaction i + T — » p + R, two cross sections 
may be similarly connected. In general, 

viTPi 2 = <r P RP P 2 
where p; and p p are respectively the initial and final relative momenta. 



Sec. 4B] Nuclear Level Widths 49 

In (33) we could have defined T pa and p c somewhat freely. They 
might contain averages over all levels in a narrow band of energy, or 
they might be restricted to levels of a single class, e.g., only those for 
which the angular momentum involved could in fact be given to the 
particle p. Since the level densities are not accurately known, such 
distinctions are ordinarily not carefully made. 

B. Calculation of Level Widths 

The assumption of the independent decay of the compound nucleus 
can be exploited in detail for the calculation of level widths. With this 
idea we combine the evident fact that the forces on nuclear particles 
which operate at large distances from the nucleus — distances large com- 
pared to the range of nuclear forces — are, we believe, completely known. 
These forces are, of course, (1) the very important Coulomb force on 
all charged particles, and (2) the effect of angular momentum, the 
centrifugal force associated with non-vanishing angular momentum. 
Most of the success of the statistical theory really comes from the fact 
that nuclear reactions are greatly affected by these essentially non- 
nuclear forces, and separation and calculation of their consequences is 
itself a considerable step toward explanation of the most evident phe- 
nomena of the reactions. 

We shall show that under suitable conditions the partial width for 
emission of a definite particle with given orbital angular momentum 
I and energy E P) leaving behind a particular state of the residual nucleus, 
can be written as the product of three factors: 

T p ia(W n ) = ky(n)P pl (Ep) (34a) 

where k is mv/h. for particle p, y(n) is a factor which depends only upon 
the state of the compound nucleus, particularly its excitation energy 
W n , and P P i is the usual coefficient of penetrability of whatever barrier 
the particle encounters from electrostatic or centrifugal forces. 

Let us begin by writing the total width of a given state of the com- 
pound nucleus, which we shall index by T(n), as a sum over the partial 
widths, one for emission of each kind of particle indexed by i, with angu- 
lar momentum Z, leaving the residual nucleus in state a. [Compare 
(33) and (34a, b).] This implies that the wave equation H^ n = W n ^ n 
for the whole system has been solved by the wave function V n = 2p a fo, 
where by i we index all possible emitted particles, for which ^ } - is the wave 
function outside the nucleus, and <p a is the wave function of the corre- 
sponding residual nucleus in state a. We obtain a set of discrete but 
widened levels by imposing this condition on the solution, with the 
specification that the function \pi contain only outgoing and no incoming 



50 A Survey of Nuclear Reactions [Pt. VI 

spherical waves. Below the energy for which some particle can be 
emitted, the function ^; will correspond to a bound state for that particle; 
above the threshold, to an outgoing spherical wave. It may be noted 
that, just as the presence of non-zero T(n) means that the states of the 
compound nucleus are broadened, and that their amplitude decays, so 
the complex W„ = E n — iT n /2 results in a complex wave number k p 
for the outgoing particle. The amplitude of the outgoing particle wave 
function is greater at large distances, since the particles represented as 
farthest from the nucleus were emitted earliest, when the compound 
nucleus had its largest amplitude. The asymptotic expression for the 
radial part of an outgoing wave e ikr /r will be damped by the presence of 
an imaginary term in k. Below the threshold for emission, this is the 
only term. Above the threshold, we can neglect this imaginary part so 
long as the width T n is small compared to the kinetic energy of the 
emitted particle. With this in mind, we write 

Yi m (6,<p)R pl (r p ) 
*p = 2-, 

l,m r v 

where R p i/r p is the radial part of the wave function of particle p, and 
Y e m is the normalized spherical harmonic. 

1. Barrier Penetration. The value of a partial width T p i{n) is just 
h/rj,;, and hence just h times the number of particles emitted per second 
from one compound nucleus. We normalize R p i by the condition that 

I | 'Apa \ 2 dV — 1, where the integral is extended over the spatial 

i/nucl 

region "within the nucleus," say out to a fixed distance R. This implies 
that T P i is small, so that we can neglect the more remote part of the 
wave function of the compound state. This is closely related to our 
general assumptions. Then the width is just h times the outgoing flux 
of particles integrated over a distant sphere of radius r x . We obtain 

r h 2 

T p i(W„) = hrj- v x \ f pl {r x ) | 2 <Kl = — • k\ R P i | 2 (34b) 
J m p 

We have required that very far away, beyond a radius r x , the radial 
dependence of R p i be simply e z ^ kr+s) . This means that we have chosen 
r x so large that in that region the kinetic energy h. 2 k 2 /2m is very large 
compared to the electrostatic or centrifugal barriers. From this point 
on, the flux of outgoing particles falls off only geometrically, with 1/r 2 . 
Hence the width becomes independent of r. But, in the region between 
the radius R which marks the boundary of the compound nucleus, and 
r w , which is the force-free region, the behavior of the wave function is 



Sec. 4B] Nuclear Level Widths 51 

fully known. It is attenuated by the requirement of penetration through 
a potential barrier, which may be zero for neutrons of zero orbital angu- 
lar momentum, but would be very considerable for alpha-particles of 
any orbital angular momentum if the kinetic energy were not high com- 
pared to the Coulomb energy, and would be appreciable even for neutral 
particles of non-zero angular momentum. We shall write R p i(r x ) in 
terms of the value of R P i{R) at the nuclear surface R. The quantity 
P p i, which we call the penetrability of the barrier, we define as the 
fraction of the initial intensity of particles, of the type indexed by p and 
angular momentum I, with kinetic energy E, which penetrates to a 
field-free region. We can write the Schrodinger equation in the field- 
containing region for the radial variation only: 

\ d 2 2m P/ 1(1+1)1 

If the solution F pt is that which corresponds to a unit outgoing wave, 
with asymptotic value e +i( - kr+i) , then evidently 

_ external intensity 1 

intensity at nuclear surface | F v i(R) \ 2 

For uncharged particles the solutions F pi are well known to be the 
half-integer-order Bessel functions. The boundary conditions are satis- 
fied by the so-called Hankel function of the first kind, 



M?r 



H l+ y 2 w (kr) 



A few penetrabilities for uncharged particles of various angular momenta 
can be given: 



Pnfl = 1 Pn.l = — — "T P, 



x 2 ~ 4 



n,2 



1 + x 2 "■' x * + 3x 2 + 9 

x 6 

n ' 3 ~ ?+te* + 4& a + 225 (36a) 

with x = kR. And, for general I values, if the wavelength is long or I 
large, i.e., if x — kR <JC I 

= x» 

n ' 1 ~ (21 - l)\2l ~ 3) 2 (2l - 5) 2 • • • l 2 (36b) 

A table of such functions has been published (Ul). 

The calculation of Coulomb barriers is much more difficult, because 
of the well-known slow convergence of the asymptotic expressions for 



52 A Survey of Nuclear Reactions [Pt. VI 

Coulomb wave functions. There are two approaches : an exact calcula- 
tion can be made by using the published tables of Coulomb wave func- 
tions (Y5). Such a calculation has been made and graphs published 
for exact penetrabilities for protons for various angular momenta emitted 
from the lightest nuclei, from Li to F (C5). The other method is the 
familiar approximate solution for the wave equation in the classical 
limit. This is the W.K.B. method, useful under conditions in which 
the distance traversed by the particle while its wavelength changes appre- 
ciably is many wavelengths. If the usual graph of the energy barrier is 
recalled, it will be seen that the approximation is good only when the 
energy of the particle does not come near the top of the barrier. This 
means that the W.K.B. method is useful either in the trivial case far 
above the barrier when the penetrability is evidently unity, or in the 
very useful case of low penetrability. In the transition region, for P of 
the order of unity, the W.K.B. approximation is not reliable. The 
relevant formulas will be given here, though they are common in the 
literature. 

The Coulomb penetrability from the W.K.B. approximation can be 
written 

Ppi(E) = ^— - 1) exp (-2C,) (37a) 

Here Bi is the barrier height for a particle of charge ze, reduced mass 
m — nipM/nip + M, and angular momentum I; Ze is the nuclear charge 
and R the nuclear radius. The quantity Ci is the familiar phase integral 



/2m\ H r r > 



Ci = [ — ) J (V-E)^dr (37b) 



where the potential energy, including Coulomb and centrifugal terms, is 

zZe 2 1(1 + l)h 2 

V(r) = + ~- and V(r e ) - E = 0; B t = V(R) for any I 

r 2mr 

The value for Ci is complicated: 

Ci 1 /tt .1-2* 



1 Ar 1 - 2» \ 

— = — I — f- arc sin — 1 — (1 + y — x) 

C A 2(x)^\2 (1 + 431/)*/ 



H 



/2zZe 2 RmV A 2(x) H \2 (1 + Axy)' 

(1+2(2/)%* +(1 + 2/ -x)*]) 

+ (y) log ^Ti^ (37c) 

with y = 1(1+ l)/(2zZe 2 Rm/h 2 ), and x = E/B t . A plot of Ci has 
been published for a sizable range of x and y; when used with the other 



Sec. 4B] 



Nuclear Level Widths 



53 



graphs in Appendix I, it makes the necessarily long computations as 
convenient as can be expected. 

It is plain from formulas (37) that the orbital angular momentum of 
the partial wave concerned plays a large part in determining the ampli- 
tude at the nuclear surface. High orbital momenta mean higher centrifu- 
gal barriers, and the penetrability through the centrifugal barrier may 
be small for high I, even when the simple Coulomb barrier has been sur- 
mounted. It is useful to plot a few barrier heights to show where this 
approximation may be expected to be satisfactory. We assume the usual 
formula R = r A* with r = 1.5 X 10 -13 cm. 

TABLE 3 
Barrier Heights (in Mev) tor Various Particles and Angular Momenta f 



Nucleus 


Angular 














Particle 


Mo- 


Ne 10 20 


Ca 2 „ 40 


Zn 30 66 


Sn 60 112 


Yb 70 m 


Th 90 232 


Emitted 


menta 














V 





3.5 


5.6 


7.1 


10.0 


12.1 


14.1 




1 


6.1 


7.2 


8.3 


10.8 


12.7 


14.5 




2 


11.1 


10.4 


10.6 


12.4 


13.8 


15.5 




3 


18.7 


15.2 


14.0 


14.8 


15.6 


17.0 


d 





3.5 


5.6 


7.1 


10.0 


12.1 


14.1 




1 


4.8 


6.4 


7.7 


10.4 


12.4 


14.3 




2 


7.3 


8.0 


8.8 


11.1 


13.0 


14.8 




3 


11.1 


10.4 


10.6 


12.3 


13.9 


15.5 


a 





7.1 


11.2 


14.3 


20.0 


24.1 


28.1 




1 


7.7 


11.6 


14.6 


20.1 


24.3 


28.2 




2 


9.0 


12.4 


15.2 


20.5 


24.5 


28.5 




3 


10.9 


13.6 


16.0 


21.1 


25.0 


28.8 



t These barrier heights were computed by using R = 1.5 X 10 -13 A 1 ^ cm. 

Now we can collect the results of the last paragraphs. Using (34b) 
and (35), we can write 

h 2 



T P i(W) = — -*-| R p i(R) \ 2 -P p i{E) 



(38) 



We identify this expression with our formula (34a), if we set for the 
factor G 

h 2 



m v 



R P i(R) 



(39) 



54 A Survey of Nuclear Reactions [Pt. VI 

It is evident that the effect of the external barrier is contained wholly 
in Ppi) we can call the product ky p i the "width without barrier." The 
argument of the function R p i contains explicitly the nuclear radius, and 
the function represents the motion of the emitted particle within the . 
reach of the specific nuclear forces. It is clearly a function of the internal 
state of the compound nucleus, as we had expected for y. Strictly 
speaking, it will also depend on the wave function and its derivatives 
outside the nucleus, on the kind of particle, and so on, for the function 
R P i must be a continuous solution of the wave equation. It is the point 
of the statistical treatment to ignore the dependence of 7 on the external 
part of the wave function. Especially if we remember that the nuclear 
forces are strong, that the radius R is rather well-defined though not 
sharply so, and that we should try to apply the theory to the average 
behavior of levels and not to specific ones, will we accept this assertion. 
A more formal argument has been given, but it cannot be made air- 
tight because of the very nature of the theory. It seems more satis- 
factory to accept this simplification in the spirit of the main Bohr 
assumption: that the compound nucleus decays in ways independent 
of its mode of formation and of the details of single levels. 

SECTION 5. THE COURSE OF NUCLEAR REACTIONS 

A. The Steps of the Reaction 

We can now proceed to the discussion of the nuclear reaction as a 
whole. The process of formation of a compound nucleus and its subse- 
quent disintegration can be described by a reaction cross section written 

this way: „ ._. t _ ,,„. 

J <j iv = Si(E)-H-D p (40) 

Here Si is a cross section for reaching the nuclear matter, a process sup- 
posed to depend only on extra-nuclear forces; £ is a number of the order 
of unity which is called the sticking probability. It is roughly the 
probability that the incident particle will enter to form the compound 
nucleus once it has reached the nuclear radius. The factor f will carry 
our uncertainty about the validity of the statistical assumptions; it is 
in this factor that we can throw all the features about the phase relation- 
ship of the possible decay products which form part of the wave function 
of the compound state. If the nucleus is excited only at a "hot spot," 
where the lifetime of the whole compound nucleus is long compared to 
the time of possible emission of particles from the locally excited region 
of only a few nucleons, we expect the sticking probability to decrease. 
In general, the more nearly independent of energy and exact reaction % is, 
the more satisfactory is the statistical model. 



Sec. 5B] The Course of Nuclear Reactions 55 

We have written as a separate factor, again in the spirit of our ruling 
assumptions, the probability D p of disintegration of the compound 
nucleus, once formed, into the particle p. This can, of course, be 
written in terms of the various partial widths: 

Dp = tj< (41) 

where the denominator contains a term for all the possible products of 
the disintegration of the compound state. The possibility of alternative 
reactions, and their relationships as the incident particle type and energy 
vary, is evidently expressed by the changing magnitudes of the various 
T's. 

B. The Contact Cross Section 

1. Elastic Scattering and Diffraction Effects. The value of <j pl will, of 
course, not really be a smooth function of energy. It will reflect, through 
a variation in D p , the presence of resonances where they are well defined. 
For energies high enough so that the classical idea of the collision can 
be applied, but not so high that the collision times for bound nucleons 
are long compared to the time of interaction of the incoming particle, 
the simplest picture of the compound nucleus would lead us to think of £ 
as approaching unity: every particle that comes to the surface of the 
nucleus — as well as that can be defined — will stick to form the compound 
state. We should like to compute the factor Si by referring as completely 
as possible to the region outside the nuclear volume. This is of course 
not rigorously possible. The incoming wave which represents the inci- 
dent particle will be modified by the presence of the nucleus in many 
ways. Those scattered components which are not coherent with the 
incoming wave, but which represent inelastic scattering or the occur- 
rence of a genuine nuclear reaction, can all be referred to events within 
the compound nucleus. There are scattered coherent waves which 
represent the actual formation of a compound nucleus, with the subse- 
quent chance emission of a particle of the original energy and type. 
Much more important coherent scattered components, however, come 
from the effects of the extranuclear forces, from the diffraction by the 
geometrical surface of the nucleus, and from the "reflection" by the 
nuclear potential. These cannot be distinguished physically in any way 
from the elastic scattering following compound nucleus formation. The 
inverse process alone provides a means for evaluating the probability 
of such elastic scattering by compound nucleus formation, and thus a 
precise definition for <S,. 



56 A Survey of Nuclear Reactions [Pt. VI 

Let us write the usual expression 1 for the wave function of an incident 
plane wave of particles of wave number k = mv/h., where m is the mass 
and v the velocity of the incoming beam in the center-of-mass rest 
system. This expression is a sum of partial waves written in spherical 
coordinates, each corresponding to a given value of the orbital angular 
momentum. We normalize to an incoming flux of unity by writing 

* = j-u **" = —^u Z (21 + l)^-i l -J l+ y 2 {kr)Y l \e) (42a) 
(v) /2 (2krv) /2 1 =0 

Here Ji+y 2 is the Bessel function, and Y° the normalized spherical 
harmonic. At great distances from the scattering center we obtain 

f _* (lY . 1 J2 (21 + l)X.il.( e -«*r+lW2) _ e -N(*r+I,/2) )Fj (42b) 

\tv kr o 

This is in the absence of any effect of the scatterer. But the scatterer 
may change in general the phase and amplitude of the outgoing part 
of the wave at large distances. Suppose that the relative phase and 
amplitude of the outgoing wave after scattering is given by the complex 
number 77. Clearly | 17 | is at most 1. Then the wave function in the 
presence of the scatterer is no longer given by (42b) but by a sum of 
incoming and outgoing waves proportional to e~ lkr and e +lkr respec- 
tively. The incoming portion is 

fcn = S*in (,) , *in W - (-)" ^ (2* + 1) *6-«*'+"' 2 > F,<°> (43a) 

\v/ kr 
and the outgoing 

^out = 2>/w W , *out (I) = ( T X(- (21 + l)*(+„)e+«*'+"'»r I < , » 

\v/ kr 

(43b) 

For a particular value of I the number of particles absorbed per second 
is just the difference between incoming and outgoing flux evaluated over 
a large sphere: 

r 2 f[v\ hn w ! 2 - v\ +oJ l) | 2 ] dSl = ^ (21 + 1)(1 - | „ | 2 ) (44a) 

Since we normalized the incoming wave to unit flux in the beam, the 
cross section for absorption is given by 

•Tabs = 2ff ab . (,) <7 ab s W = J (21 + 1)(1 - I i, | 2 ) (44b) 

1 This whole treatment follows closely that of (F7). 



Sec. 5B] The Course of Nuclear Reactions 57 

The elastically scattered wave has to be added to a plane wave in 
order to get the actual perturbed wave ^ in + ^ out . This is simply 
lAin + i^out — e* 70) H - Again the flux in this wave integrated over a 
large sphere gives the elastic scattering cross section: 

a ei = W° ffel « = ^ (21 + 1)|(1 + v )\ 2 (44c) 

It is valuable to consider the relations between these waves. The 
total cross section <r tot = 2<r tot w can be written: 

<rtot W = <r e l (0 + <r abs W =—(21+ 1)[1 + jfcfo)] (45) 

/c 

There is a well-known and interesting result which follows from this 

formula. Consider the cross section of an obstacle in the classical limit, 

with the radius (R) of the scatterer very large compared to X = 1/k. 

We expect the classical absorption cross section, summed of course 

over all values of I to obtain the total cross section, to be wR 2 if the object 

is "black" i.e., if it absorbs every particle whose trajectory strikes it. 

But in this case v = 0, and the total cross section <r tot = 2wR 2 . Where 

does the additional ttR 2 of elastic cross section arise? This is the familiar 

"shadow scattering," now observed repeatedly for neutrons of 90 Mev 

and less on heavy nuclei (CIO). The black sphere clearly casts a shadow 

m the beam. This shadow must, on the wave picture, as in the familiar 

Kirchhoff treatment of physical optics, be produced as the result of 

interference between the incident wave and a scattered wave. The 

total intensity of scattering, to produce a shadow the size of the object, 

evidently corresponds to a cross section just equal to the geometrical 

one, tR 2 . Thus there must be an elastic cross section of irR 2 . This 

can be observed because, especially in the nuclear case, the shadow does 

not extend to infinity even in a parallel incident beam. On the contrary, 

diffraction by the sphere means that the direction of the scattered wave 

is changed by an angle of the order of X/R. This has the consequence 

that the shadow is dissolved by the diffracted beam in a distance of 

R 2 /X from the scattering center, and the elastic scattering can there 

be observed. 

The angular distribution of the shadow scattering is easy to compute 
for a black sphere, in the limit X/R « 1. The result is that 

da(8, <p) _ R 2 J x 2 (kR sin 8) 

da ~ ~ shTf? (46) • 

where k = 1/X = h/M v , R is the nuclear radius, and <p the scattering 



58 A Survey of Nuclear Reactions [Pt. VI 

angles, and.Ji(z) the Bessel function of the order of unity. Actually the 
contributions of partial waves with I ~ R/X are small but complicated. 
If K/R ~ 1, the effect of important values of I will not be given correctly 
by (46), and the distribution and magnitude of the elastic shadow scatter- 
ing will be more complicated. The success of this result has suggested a 
nuclear model of optical type in which the nuclear volume is regarded 
as a sphere of "gray," not black, material which both absorbs the inci- 
dent wave in part and disperses it, as a result of the phase shift due to 
the mean potential which the incident nucleon feels in passage. This 
model has given interesting results in the high-energy region, which is 
discussed in more detail in Section 11. 

It is worth pointing out that the maximum value of Caba^ is (21+ l)wX 2 , 
and that this maximum can be reached only if y = 0. The maximum 
value of <r e \ w is (21 + l)47rX 2 and can be attained only when y = +1. 
There is in general a range of values of tr e i (i) possible for each value of 
(Tabs^. That the maximum scattering cross section is four times the 
maximum capture cross section reflects the fact that maximum scattering 
comes from just reversing the phase of the incoming wave to form the 
scattered outgoing wave. This has the effect of adding to the plane 
wave just twice its outgoing part, while complete absorption simply 
removes the outgoing portion of the plane wave. Thus the cross section 
for scattering depends on the square of twice the amplitude correspond- 
ing to complete absorption, which gives four times the cross section. 

2. Influence of Extranuclear Forces. We have so far considered only 
maximum and minimum values for scattering and absorption, obtained 
by assuming values for the quantity -r\. It is clear that a real calculation 
of 17 would require the solution of the many-body nuclear problem, 1 
and this is precisely what we are unable to do. But we can try to bring 
explicitly into view the effect of extranuclear forces, reserving £ to de- 
scribe the effect of the specific nuclear interaction and the formation 
of the compound state. 

The nucleus, in absorbing the incident beam, changes the intensity 
of the outgoing portion of the wave. If there is no absorption, J 77 j = 1 
and we can write i\i = e 2is l and | fe | 2 = | y out | 2 . With absorption, 
some of this wave is removed, and we can write | ^ out | 2 = | Win | 2 
— I ^ remove d | 2 . But we will regard the wave removed as altered by the 
nucleus proper only at the nuclear surface, writing then : 

Wout I 2 = I Win | 2 - f\ ^urf | 2 = I Win | 2 - f\ Win | 2 ■ ^^ (47) 

I Win I 

1 An important paper (W7) is based on an effort to throw the whole burden of 
describing the reaction onto the quantity jj. We shall discuss this in Section 6. 



Sec. 5C] The Course of Nuclear Reactions 59 

with the quantity / representing that fraction of vW which is removed 
by the effect of the specific nuclear forces. But we have already shown, 
in (35), that we could write | ^ surf j 2 /| ^ in | 2 = P. With Eqs. (44) in 
mind, we get 

I *o»t I 2 = I * ia | 2 (1 - fP) a abs ~ U in | 2 - I ^ out | 2 ~f P \ fc„ J2 

(48) 

where we can regard f as an expression of the sticking probability £, 
discussed above. This would allow us to extend the notion of the stick- 
ing probability, as some authors have done, even to reactions where the 
statistical notions were not strictly applicable, in order to give a kind of 
transition between statistical and other views. We then can take as a 
fundamental relation for the cross section for compound nucleus for- 
mation : 

*,-,= (2Z+l>*.- 3 -Ptf(^)-&i (49) 

with the barrier penetrability and sticking probability explicitly indi- 
cated. 

C. The Disintegration of Compound States 

1. Competition. The next step in the nuclear reaction, the disinte- 
gration of the compound state, takes place with a probability D p = 
r p/ 2l V (41), as we have already said. We ought to recall here the na- 
ture of the assumption behind this way of writing the reaction: we 
assume that particular features of any single compound level are not 
important. Many states are involved, with random phase relations, 
either because of the poor definition of the particle energy in the incident 
beam, or because the states are so widened as to overlap. Otherwise, 
the results we give will be valid only for averages over many states or 
even over many similar but not identical nuclear species. 

The most striking consequence of the expression (41), for D p is the 
phenomenon of competition. Evidently once the compound state is 
formed it can decay in many ways, for each of which there is a value of 
the appropriate T,-. The energy of the compound state will determine 
the value of the r/s in our approximation at least. As the energy 
varies, given reactants may produce different products. As each new 
energy threshold for particle emission is passed, the corresponding r,- 
rises from zero, and the compound state has then another possible decay 
mode. Since the total cross section is limited, the rise of one Tj must 
reduce the yield for the others. This is called competition. For a 



60 A Survey of Nuclear Reactions [Pt. VI 

definite example, consider the reactions induced by neutrons incident on 
Br 81 . The reactions which have been observed are four (S5) : 

(i) Br 81 (n, 7 )Br 82 

(ii) Br 81 («,p)Se 81 

(hi) Br 81 (n,a)As 78 

(iv) Br 81 (n,2n)Br 80 

At very low incident energy (thermal neutrons) only T y is different from 
zero, and the first reaction takes place exclusively. As the energy in- 
creases, the threshold for the second reaction is reached at a couple of 
Mev, and it begins to appear. At still higher energies the latter two 
reactions are possible, and their competition is noticeable in the cross 
section of the (n,p) reaction. It is clear that the computation of <r„ for 
the formation of the compound nucleus will apply to all these reactions 
(though the resonance neutron capture will need special treatment) and 
that the values of the individual cross sections can be obtained if only 
the relative T's are known. 

2. Specific Level Widths for Particle Emission. We have already de- 
scribed, in Eq. (33), how the widths are related to cross sections, and 
we have given formulas for the computation of individual widths. In 
practice, of course, we want not so much the width for emission, say, of 
the proton in reaction (i) above, leaving the Se 81 nucleus in a given state, 
but rather the total width for the entire range of proton energies and 
residual nuclear states which are available. Of course, we have but to 
build this up out of the various individual widths, and we shall obtain 
at the same time the often interesting energy distribution of the out- 
going protons. 

From (33) we have the value of T pa for the emission of a given particle 
with definite energy (averaged over many levels). If now the residual 
nucleus can be left in many possible states a, so close that they too can 
be represented by the statistical expression for the level density, we ob- 
tain, for the total width for particle p, T p = ^ T p i a , which we can write 

l, a 

in favorable cases as an integral: 

1 /»Ej, max m fi 

— - — • ~-j<7 p {E)p R (E™**-E p )dE p (50a) 



r P = 



Here all factors are known: a p is obtained from suitable use of (49), 
the level densities are given by some model, taken say from Eqs. (31), 
and only the factor £ which occurs in <r p expresses the model's lack of 



Sec. 5C] The Course of Nuclear Reactions 61 

preciseness. (We suppress the weight g R , including it in the value of PR; 
the level density.) We hope to find the £ variation small, and the value 
of £ near unity except in special cases. Thus f p can be computed for 
charged nuclear products or for neutrons. We shall have to reserve 
the gamma-ray width calculation for a later section; it is almost always 
small compared to heavy-particle widths as soon as the emission of 
heavy particles is energetically allowed. For the computation of D p 
clearly the factor l/ Pc (W) is contained in all terms r y , and the result is 
a function of E p maK only. The dimensionless integral of (50a), without 
the factor \/t? Pc , is referred to in the literature (W6) as the / P (-Ep max ) 
for a given state (S5). Under specific headings we shall discuss the 
calculation of these /'s in more detail: 

/ P (^ max ) = ir 2 Pc(W)T p (50b) 

3. The Thermal Analogy: Temperature, Cooling, Evaporation. From 
the definition of the function f p it is seen that f p is just the width for 
emission of particle p from the excited compound nucleus, measured in 
terms of the mean spacing of levels at that excitation of the compound 
nucleus. There is in addition the numerical factor w 2 . For this reason 
we shall refer to the dimensionless / function as the specific width for 
emission of particle p. 

It is interesting to view the process of emission of particles from the 
compound state as a kind of thermal evaporation. Just as a water mole- 
cule may evaporate from a drop of water, so does one of these nucleons 
leave the excited nucleus, in which the excitation energy plays the role 
of thermal agitation. Equation (50a) leads to an expression for the 
energy distribution of the emitted particles, which is of course just the 
integrand in the expression for T p . We have for I(E P ) dE p the relative 
number of particles p emitted with energy E p in the interval dE p : 

I(E P ) dE p = I E p <t pPr (E™* - E p ) dE p (51) 

where the constants have been lumped into 7 . Now from Eq. (30) we 
can write the level density PB in terms of the entropy of the nucleus, S. 
We get 

_ IpEpffp exp [SnjEjT* - E p )] 

T(2irdW B /dT) y * 

Now we shall make an approximation which is justifiable only for 
sufficiently high values of the excitation energy of the residual nucleus 
(£p max - E p ). We shall expand the entropy about the maximum value 



62 



A Survey of Nuclear Reactions 



[Pt. VI 



of excitation energy which can be left behind in the residual nucleus, 
namely about E p ma *. We obtain then 

dS 

S(W B ) = S(E™* - E) = S(E p ma *) - E p ■ 

dE p _ r 

Now, remembering that the exponential varies much more rapidly 
than the factor l/(dWR/dr)^, we use the familiar thermodynamic 
relation dS/dE p = 1/r. Inserting this, we get 



(IE P ) = const X Ej 



p<r p expl I 



(52) 



Here r is the temperature (in energy units) at which the excitation energy 



of the residual nucleus is on the average Wr = E p 



If the function 



^ 






a 






fc) 








\ Neutrons 




c 






1) 






S 








N. Protons x 10 


<D 






> 












'+3 


"/ s ^^^ 




£ 


/ / ^^»^ 






i ^ ^^^k. 




~3 


/ ^*r ^^ 




« 


' ^^ i i 


' ^> 



4 8 12 16 

E p , kinetic energy of emitted particle (Mev) 

Fig. 6. Energy spectra of evaporated nucleons. Note reduction of proton 
evaporation due to Coulomb barrier. 



Epff p is not too rapidly varying, as it is not for neutrons of some Mev 
energy, for example, the emitted particles have exactly the Maxwell 
distribution in energy, just as do the molecules evaporating from a drop. 
The temperature T corresponds to the temperature of the drop after 
emission of the particles — the temperature of the residual nucleus — 
which is not the same as that of the compound state, for of course the 
single emitted particle has a non-negligible fraction of the excitation 
energy. This makes the evaporation analogue somewhat less exact; 
we must think of a drop with only a hundred water molecules. If the 
particles p are charged, the cross section will be quite energy-sensitive; 
low values of E p will mean that the penetrability of the barrier is low, 
and will distort the Maxwell distribution, shifting the maximum to high 
energies (see Fig. 6). 



Sec. 5C] The Course of Nuclear Reactions 63 

If the variation of a p with E p is neglected, the maximum energy of 
emitted neutrons will lie at E p = r. Table 4 gives temperature as a 
function of excitation energy for several nuclei; these temperatures were 
obtained by using the level density expression of (31). That they are 
clearly rather small compared to the excitation energy helps justify 
the approximations made. This means also that the outgoing particles 

TABLE 4 
Nuclear Temperature in Mev, Atomic Number and Excitation Energy 
A- = 80 120 160 200 240 



Excitation 












Energy (Mev) 












2 


0.61 


0.51 


0.46 


0.43 


0.41 


5 


0.96 


0.81 


0.73 


0.68 


0.65 


10 


1.36 


1.15 


1.04 


0.96 


0.91 


20 


1.93 


1.62 


1.47 


1.36 


1.29 



will in general take away only a small part of the excitation energy, 
leaving behind an excited residual nucleus, which may in turn boil off 
still another particle, and so on. It is this process of gradual "cooling" 
by "evaporation" which is often the origin of the complicated multiple 
processes listed in Appendix II. 1 

It is not easy to justify the two assumptions: (a) the statistical treat- 
ment of the level density, and (b) the neglect of E p compared to E p max . 
The level density formulas certainly overestimate the density of the 
lowest levels, or at least the lowest ones accessible from a compound 
state experimentally defined. Certainly the formula will not be very 
reliable below E p m »* of 5 Mev, and for # p max - E p of less than a couple 
of Mev. Above that it should be fairly reliable, until the excitation 
energy reaches values of 10 Mev or more per particle. The lightest 
nuclei will of course make difficulties for the statistical treatment. 

The release of neutrons from the moving fission fragments by evapora- 
tion is to be expected and should account for the prompt fission neutrons, 
and a spectacular example of "boiling off" is found in the high-energy 
fission of bismuth, where apparently the most probable fission fragments 
indicate that the compound nucleus which divides by fission is 12 mass 
units lighter than the state initially formed. This implies the loss of 
many neutrons by something very like the evaporation we have de- 
scribed. The nuclear "stars" of cosmic rays often represent such proc- 
esses. We shall discuss them at greater length in Section 11D1. 
1 See, for example, (T5). 



64 A Survey of Nuclear Reactions [Pt. VI 

SECTION 6. THE DISPERSION THEORY: 
RESONANCE REACTIONS 

The statistical theory which we have described has been based on an 
avoidance of reactions in which the properties of a single compound 
state are prominent. But some of the best-known and most interesting 
nuclear reactions are exactly those in which spectacular resonances 
indicate the importance of the properties of the individual states. 
Starting from the idea that such levels must be considered in detail as 
individuals, several authors (B13, B21, K2, S9) have tried to find a 
general quantum-mechanical solution of the many-body problem pre- 
sented. They based their treatment on the same two-step notion of the 
reaction as does the statistical treatment of Bohr. The compound 
nucleus is the system of incident particle plus target, a system whose 
eigenvalues are not real and whose eigenstates are not stationary but 
widened and quasi-stationary, as we have described them. The forces 
of interaction between incoming particle and target cause transitions 
between the initial state of the incident particle and an intermediate 
state which is one of the widened states of the whole system. Then this 
state, by virtue of the internucleon and nucleon-radiation perturbing 
couplings, itself decays to the state which contains outgoing particle 
and residual nucleus. This description was called the "dispersion 
theory" because of its formal near-identity to the calculation of the 
optical dispersion of atoms, in which the incident quanta are first 
absorbed by the atom, and then another quantum, perhaps the same in 
energy as the first, is re-emitted by the excited atom. The initial and 
final states of the nuclear system, then, are combined only by the mixing 
through the compound state, and not with any considerable probability 
directly. This is the consequence and of course the motivation of the 
idea of the compound nucleus. The "mixing" is strongest when the 
energy of the initial system coincides with some more or less well-defined 
energy level of the compound state. It is of course the intention of this 
form of theory to give a complete account of nuclear reactions, and the 
results of statistical theory are expected to follow from dispersion theory 
calculations when suitable averages are taken, and suitable assumptions 
made about the relative phases of the wave functions involved and 
about the character of the levels of the compound state. Whenever 
really sharply defined incident energies are experimentally available, 
and whenever the states of the compound nucleus are reasonably well- 
defined, the dispersion theory ought to yield more information than the 
statistical model, however improved. 



Sec. 6A] The Dispersion Theory: Resonance Reactions 65 

The program of the dispersion theory is too ambitious for full success. 
The method of perturbation theory which works well for the weak 
radiation coupling of the electrons in an atom cannot be expected to 
give, even in theory, an adequate scheme for the calculation of the result 
of the strong interactions among nucleons. 

Wigner and co-workers have given a beautiful general theory (see 
Appendix II), if a rather complicated one, which does not employ the 
ideas of perturbation theory but insists only that the nuclear forces act 
within a well-defined and not too large region of space. Using the ideas 
of the ordinary Schrodinger wave equation, we shall indicate the physical 
connection between a simple one-body model of nuclear reactions and 
the most useful results of the complete dispersion theory, but for full 
discussion of Wigner's S-matrix treatment the literature should be 
consulted. 

A. The One-Body Model and Its Difficulties 

The principal notions of nuclear reactions before the early 1930's 
were based on a model much simpler than the one we have been dis- 
cussing. It was built up largely from a study of the decay of alpha- 
radioactive nuclei and the light nuclear resonance reactions with alpha- 
particles. The picture was that of the Hartree model of the nucleus, i.e., 
that the many nucleons produced a net potential well, a combined force 
field in which the particle to be studied moved. In alpha-decay the 
alpha-particle shuttled back and forth in this well, until, once in a while, 
by chance it leaked out of the barrier by the familiar "tunnel effect" of 
quantum mechanics. For scattering, the nuclear forces supplied a 
potential well in the same way. To account for the resonances observed, 
the charged particle was thought of as penetrating the external barrier 
and moving into the potential well; when the particle wavelength was 
just such that the particle could produce a standing wave in the well 
by virtue of in-phase multiple reflections, the particle energy was in 
resonance with one of its possible stationary states in the potential well. 
Constructive interference built up the wave function strongly, and 
absorption grew very rapidly at several well-defined energy levels. 
This worked quite well for such reactions as Mg 24 (a,p)Al 27 . It seemed 
indeed to be a reasonable model. The first slow-neutron resonance 
work showed up its major weaknesses. It became evident that in this 
theory scattering and absorption cross sections should increase together. 
Barring unusual selection rules, the increased particle amplitude inside 
the well should lead to increased re-emission — scattering — no less than 
to increased absorption. Even if capture were somehow prevented for 
some scattering resonances by peculiar effects, which might limit the 



66 



A Survey of Nuclear Reactions 



[Pt. VI 



dropping down of the particle through radiation to a lower level in the 
potential well, it seemed hard to understand why every level which 
permitted capture would not yield an even stronger resonance scattering. 
The simple analogue of the absorption of sound of the resonant frequency 
by a Helmholtz resonator is complete. The constructive interference 
of the wave reflected within the resonant cavity builds up its amplitude; 
sound energy is not only absorbed, but also strongly re-radiated. Yet 
experiment showed that strong absorption is not accompanied in general 
by strong scattering. 

We have already formulated the general problem of scattering and 
absorption by a nuclear center of force. Let us inquire into the results 
of a one-body model, using this formalism (Eqs. 41 et seq.). Consider 
the case appropriate for thermal neutrons, with wavelengths very large 
compared to nuclear dimensions. Then we can write for the radial part 
of the wave function in the region outside the nuclear radius R 



* = 



1 u(r) 
(ti)^~ 



«( r ) = V e ikr + e- ikr 



(53) 



Only the 1 = partial wave is important, and outside the radius R there 
are no forces. From Fig. 7 we can see that this external wave function 




r^ oat = ve ikr +e- 



-V 

Fig. 7. Wave function for scattering of thermal neutrons from nuclear potential. 

must be joined to the internal solution. For definiteness let us consider 
that the internal potential acting within the radius R is represented by 
a rectangular well. This is in no way essential for the result, but it 
simplifies calculation greatly — at the expense of realism. Within the 
nucleus, then, the neutron has the wave function 



^int = 



u(r) 



u(r) = A cos (Kr + e) 



(54) 



Sec. 6A] The Dispersion Theory: Resonance Reactions 67 

with K some wave number, much larger of course than the wave number 
* of the slow neutron outside the nucleus, and e some phase constant 
The boundary condition at the origin will demand that e = nr/2, with 
n odd. The wave equation now requires that the constants be so ad- 
justed that both ^ and ty/dr remain continuous across the nuclear 
boundary. Let us define the quantity /, the logarithmic derivative of 
the wave function at the nuclear edge made dimensionless by a multi- 
plicative constant : 

/ = R (du/dr)/u | r=K (55) 

Then the continuity conditions are both satisfied if we require / in = f outl 

(r,e ikR — P ~ikR\ 

-KR tan (KB + e ) = ikR — i 

One %kR + e~ ikR ) 
Solving for rj, we obtain 

-2ikR (#in - k R) 

Win + kR) ^ 7) 

Recall expressions (44). They give 



(56) 



ffrf » (0) = p d ~ I " I 2 ) and ,/) = J: (| i + , | 3) 



k 2 

If we find v , both cross sections are determined. Clearly we have not 
yet allowed for the possibility of absorption— we have used only one 
level within the well-and our | „ | 2 = 1. This follows from the form of 
(57) and the fact that / in is purely real. Now we make use of the small 
value of kR « 1 for slow neutrons. We can write (57) as 

_ (-if + kR)(if - kR) fc 2z-,2 

'= fTm- *" d [1 + F ^'f^ 

by expanding. Then the cross section becomes 

<r^=~[l + \v\ 2 + 2ne( v )]=^.-Jf^- 
k 2 k 2 f + k 2 R 2 

<kir 1 

~ K 2 ' tan 2 (KR + nr/2) + k 2 /K 2 (58) 
This cross section shows a maximum at zero energy regardless of the 
nuclear size. But, for nuclear sizes and internal motions such that 
KR + mr/2 = 0,7r,- • •, the small cross section at low energy becomes 
instead an infinite peak. If we fix a definite value of slow-neutron 
energy and imagine that the nuclear radius, say, varies smoothly 



68 A Survey of Nuclear Reactions [Pt. VI 

strong scattering resonances will appear for special values of the nuclear 
radius. Somewhat more familiar-looking resonances, appearing for 
definite values of incoming neutron energy, would have resulted had 
we considered partial waves of higher angular momentum, but the 
general features would not have been very different. 

Now, moreover, we can calculate the amplitude of the wave inside the 
well, taking an incident wave of unit flux. Using (54) and (56), we get 



1 o / n A 



A 2 ^ 
K 

Thus the amplitude of the neutron wave inside will become large at 
resonance, and in fact we can write the cross section for scattering in 
terms of that amplitude, at least near resonance : 

o- ac (0) = 4xA 2 near resonance 

But, clearly, the radiative transitions to a lower state will have a matrix 
element proportional to the amplitude A, and the probability of such 
transitions, and thus the cross section for absorption, will vary like A 2 , 
as we should physically expect from the fact that A 2 measures the time 
the particle spends inside the nucleus. The ratio <x a bs/o- S c will therefore 
show no marked change at resonance; strong scattering and absorption 
resonances will be found at the same energy. This is contrary to all 
experience. Furthermore, it is easily seen that the resonance levels 
are spaced in energy much too little to correspond to adding another 
half-wave to the inside wave function, and that the variation of cross 
section with nuclear radius (i.e., with the mass number A) is far too 
erratic to be accounted for on such a picture. The one-body model 
gives much better results when the external wave function determines 
the broad course of events, as in alpha-particle radioactivity. Where 
the interior state of the nucleus is decisive, the model is inadequate. 
The strong physical plausibility of the compound nucleus picture, added 
to the difficulties of the one-body description of neutron capture, gives 
it its present importance. The modifications which the strong effects 
of shell structure will certainly require, especially at moderate excitation, 
have still not been completely worked out. 

B. The Dispersion Theory for an Isolated Resonance 

The difficulties which surround the derivation of the theory of nuclear 
reactions by perturbation methods have led to reconsideration of the 
basis of the whole matter. It turns out that the most important features 
of the theory arise from the fundamental nature of scattering and ab- 
sorption processes themselves, which are displayed fully only in the 



Sec. 6B] The Dispersion Theory: Resonance Reactions 69 

rather complex nuclear domain. The general theory has been developed 
by Wigner and several of his co-workers. We shall discuss mainly a 
quite satisfactory but much simpler and less general approach developed 
by Weisskopf and others. In it the features of the one-body model which 
make scattering and absorption so pictorial are generalized to fit the 
physical picture of a compound nucleus. In the one-body model we 
have shown that, for a case when only two alternatives, elastic scattering 
and absorption, are possible, the logarithmic derivative function / 
determines the amplitude 77, and hence both cross sections. But in the 
one-body model the complex number / (which must be complex to allow 
absorption) is fixed by the nuclear potential well. Both modulus and 
phase are given. Weisskopf regards this same quantity/, now in general 
a complex number, as determined by the whole structure of the nucleus. 
It will be different from level to level, varying in a manner much more 
complicated than does the one-body /, and in particular having modulus 
and phase independently varying. We define the resonance energies 
and level widths by giving the properties of/. Thus we have a kind of 
phenomenological compound nucleus picture into which the properties 
of many levels enter, but through a single function whose determination 
from the actual make-up of the compound nucleus we leave perforce 
to the physics of the future. We shall require that (1) / is a function 
only of the energy (and other constants of the motion) of the compound 
nucleus, and not explicitly dependent on the incoming particle, and 
(2) / is a well-defined function, defined by (55), in spite of the uncer- 
tainty of the value of R, the nuclear radius, at which internal and ex- 
ternal waves are to be fitted. Both of these conditions can be met by 
the requirement that the wave function just outside the nucleus vary 
only slowly over a distance which corresponds to the mean spacing of 
the nucleons within the nuclear matter. We can regard the value of R 
as a parameter to be chosen to give the most reasonable average behavior 
of /. Some values of R, and the related internal wave number k, will 
not work. No value will allow us to assign the behavior of / uniquely, 
but the best values will yield reasonable statistical agreement with the 
observed properties of many levels. 

1. Derivation of the One-Level Formula, (a) Without External 
Forces. Guided by physical considerations, we shall now try a general- 
ization of the one-body model. The function / is defined by the ex- 
pression 

f = R (du/dr)/u \ r=R (55) 

We expect zeros of / when the slope of the wave function vanishes at the 
nuclear edge, and infinite values if the wave function itself vanishes 



70 A Survey of Nuclear Reactions [Pt. VI 

there. Without loss of generality we can take over the special form of / 
from the one-body theory: 

/= -kR tan z(W) (59) 

The tangent function no longer has as argument the simple and well- 
defined phase of the internal wave function in the one-body potential 
well, kR + «. It now has instead a function z(W), which increases 
monotonically as the excitation energy W = Ej, in< j + Eun increases, 
and takes the value nx at each of the successive resonance energies 
W r = E b + E T , but may vary as it will in between. The results of the 
theory now depend only on the assumption that the variation of z(W) 
with energy is as smooth and simple as possible. We shall see how all 
the properties of the successive resonances can be described by a suitable 
trend for z near each resonance. The prediction of the behavior of z in 
detail is given up for the present theory; it could be determined only if 
the actual motion of the entire compound nucleus in every eigenstate 
were known. 

But the one-body picture did not explicitly include absorption. We 
allow for that here in the familiar way: we introduce a damping factor 
to reduce the amplitude of the now only approximately stationary state. 
The damping arises of course out of the possibility that the system can 
change its state not simply by decomposing to re-emit the incident 
particle, but in some other way. Such a damping will as always be 
expected to widen and depress the resonance peaks. We write the 
energy as a complex quantity: W = W — zT a /2 so that the energy 
eigenfunction becomes 

Now the probability of occupation of the given state decays in time with 
the factor e~ r " t/h \ \[/(0) | 2 , and the mean lifetime of the state is given 
by r = h/r a while as usual the uncertainty principle will imply that 
the state is defined only up to a width r a ~ h/r. This will indeed follow 
from the calculated resonance shape. 

Evidently the concept of resonance level is useful only if the energy W 
has but a small imaginary part, i.e., if T a /W <K 1. Although we could 
write the function quite generally, we shall use the approximation of 
expanding / in the neighborhood of the resonance energy, leaning 
heavily upon the smooth behavior of z(W) and the expected smallness 
of T. We write then 

_ iT a df I 

f(W)=f(W)- — -= +••• 

2 dW\w=w 



Sec. 6B] The Dispersion Theory: Resonance Reactions 



71 



Taking our cue from the one-body model, let us define the resonance 
energy, E r , by the relation 

f(W r ) = f(E b + E r )=0 

and continue to expand / near the value E r . We get 



f(W) = (E - E r ) 



dj_ 
dE 



E r 



iT a df 
2 dE 



+ ■ 



(60) 



E r 



where we use the kinetic energy of the incoming particle, E, as measure 
of the excitation energy W. 

This linearization greatly simplifies our theory. It is certainly no 
serious source of error provided that we look in the near neighborhood 
of a resonance level. How far that neighborhood extends depends of 
course on the variation of the argument z(W) with energy. Now we 
can use the relation between the function / and the phase shift r?, just 
as we did in the earlier model, from Eq. (57). We take the value of / 
near resonance from (60). If we introduce explicitly the functions 
fo(E) and g(E), the real and negative imaginary parts of the function /, 
we obtain . 

f(E) = f (E) - ig(E) = (E-E r )8 + ^5 (61) 

where we have written 5 for the quantity (df/dE)\ Er , and r o = -2g/5. 
Using (57), there follows from (44) 

kRg 

(62a) 



Cabs 



(0) 



= - (1 - | , | 2 ) = 4,rX 2 



k 2 "~ '"" "" (g + kR) 2 +f 2 

We can write <r so (0) similarly. The fact that resonance is marked by 
/o = is now plain, and the whole expression gives just the familiar 
one-level Breit-Wigner formula of dispersion theory; compare (62b). 

Now, writing the cross sections out in full, but replacing the functions 
/o and g by the more physical widths from (61), and defining a width r„ 
by the relation r n = —2kR/8, analogously to (61), we have 



^abs 



(0) _ 



= xX* 



■*■ n*- a 



m 



(62b) 



+ (E - E r ) 2 



(0) _ 



= 4ttX 2 



= 4tX 2 



kR 



i(kR + g) - f 

2 1 n 



+ e lkK sin kR 



E -E r + -(T n + r„) 



+ e lkK sin k R 



72 



A Survey of Nuclear Reactions 



[Pt. VI 



From the results of (62) we can go much further in the interpretation 
of the argument function z(E) of (59). Plainly it goes through multiples 
of ir, giving zeros for the logarithmic derivative function / at each 
resonance. It is also clear that the contribution to absorption, say, of 
any level will fall off on each side of the resonance energy, with a charac- 
teristic width given by T n + T a = T. But this width is inversely 



(re+3)x 

(w + 2) 7T 

(n-hl)T 



r large 




small 



Excitation energy 



W 



Fig. 8. The argument function z(W) as a function of the excitation energy "FT in a 
region containing several resonances, W a , Wb, • • •, with differing widths (F7). 

proportional to the slope of the / function near the resonances. From 
(57) it is easy to show that 

. |2 _ 1 + a 2 Im (/) 

1 " ' 1 - a 2 Im (/) 

where a 2 is positive definite; and, since we require from the absorption 
cross-section formula (44) that | 17 | 2 < 1, it follows that Im (/) < 0, 
and in our expansion therefore 5 must be non-positive. This justifies 
our use of the minus sign in the definition of T n and r o and confirms 
the choice of the tangent function in (59). We can now see graphically 
the meaning of Fig. 8, in which the function z(E) is plotted schematically 
in a region of several resonances. The width is given in our approxima- 
tion by T = — const/5 = — const -dz/dE. Since 5 is necessarily non- 
positive, z does increase monotonically, as we expected. Where z changes 
rapidly with E near a resonance, 8 is small and the widths large. We 
have drawn narrow levels at resonance energies a, b, c, wider ones at 
d and e, and a very wide one at /. We cannot predict the level positions 



Sec. 6B] The Dispersion Theory: Resonance Reactions 73 

or the slope near those values. But we have thrown the whole burden 
of the determination of reaction and scattering cross sections into the 
behavior of a single function z, which is a kind of equivalent internal 
phase. If z varies smoothly, we can expect that dz/dE will be small, 
and the widths small when the levels are closely spaced; and reverse 
will also hold. By such simplifying assumptions on the smoothness 
and statistical regularity we can produce various results concerning the 
average behavior of nuclear resonance lines. Individual levels must be 
examined empirically, but can be fully described by this theory. The 
choice of the matohing radius R can be fixed by the pragmatic test of 
how simple and statistically unbiased is the behavior of the many levels, 
for R must divide the region outside the nucleus from the one within, 
and this division will be physically most clear cut for a particular R. 
Improper choice of R will allow the behavior of z to be affected by the 
external region and will distort the expected intelligibility of the proper- 
ties of z. 

(6) With Barrier. So far we have considered only the case appropriate 
to slow-neutron reactions, where the orbital angular momentum I = 0, 
and neither centrifugal nor Coulomb forces are present. Outside the 
nuclear radius R the potential is strictly zero. In this case, and in this 
case only, the external wave function is given not just asymptotically 
but everywhere outside the nucleus by the partial plane waves of (53). 
It is not very hard to extend the calculation to the more general case. 
We write the radial part of the wave function in the external potential 
as before, for a definite value I of the orbital angular momentum. But, 
as we approach the nucleus, the wave function is no longer a simple 
plane wave. We introduce the independent solutions of the wave 
equation with the given external potentials which go over asymptotically 
into incoming and outgoing partial plane waves. These solutions we 
call u t and u for incoming and outgoing parts, respectively. (The 
Coulomb potential can be thought of as screened very far away to 
avoid the logarithmic term in the asymptotic phase shifts.) We write, 
for unit flux, 

1 Ui(r) + V U„(r) 1 e -i^r-l,l2) + + i {k r-hl2) 

Ur) =W* r — ^ " (63) 

Now, as we come in to the matching radius R, the behavior of the solu- 
tions Ui and u is completely known for any given external potential. 
Since they are adjusted asymptotically to the same amplitude, and one 
solution could be obtained from the other by simply reversing the direc- 
tion of time, one is the complex conjugate of the other, u t = u *. We 



74 A Survey of Nuclear Reactions [Pt. VI 

shall also need the penetrability of the external barrier, which we can 
define, as in (35), by the relations 



pd) = I "^ I = i_ = i (64) 



M,(°0) J 2 _ 1 1_ 

«,(#) | 2 ~ I Ui(R) j 2 ~ Ui(R)u {R) 

Here we affix the I value corresponding to the orbital angular momentum 
partial wave involved. We shall also make use of the Wronskian rela- 
tion, obtained in the usual manner by writing the wave equation for 
each solution, cross-multiplying, and subtracting, to get 

d / du du{\ 

— I Ui u — ) = uiuj — u Ui = const 

dr \ dr dr / 

Since this is true for all values of r, we evaluate f or r — * » to obtain 
UiU ' - u Ui' = 2ik — = 2ikP (l) (65) 

U Ui 

Now let us define the quantities Fi i0 = RuiJ /ui i0 \ r= u by analogy to 
the function /. With all this not very complicated machinery, we have 
the solution at hand. Again we write down the matching condition at 
the nuclear edge, just as in (56) : 

/in = kR(u/ + rfu ')/(ui + r]U )\ r=B 

Solving for the amplitude v, we get 

Ui{R) Fi - f in Ui {R) (if ia - iA + y) 
V = = (66) 

U (R) F — /i n U {R) (if in - iA - 7) 

where we have written the expression in the form closest to (57) and have 
introduced the notation F,-, = A ± iy for the complex numbers Fi i0 . 
Writing /i n = f„ — ig as before, and evaluating y from relation (65), 

Fi - F = 2iy = -2ikRP m y = -kR-P w (67) 

we get from the fundamental relation (44) between o-„bs and 17, the result : 

±irX 2 -kRP<> l) g 

This is very like the result for the s wave obtained in (62a), and, indeed, 
for I = and no external forces, «,-,„ = e ±tkB , P (0) = 1, and A = 0, 
which gives exactly (62a). 

If we write for the unimodular number Ui(R)/u (R) = e~ 2ia , which 
was just e~ 2%kR for the no-force case, we can write the scattering cross 



Sec. 6B] The Dispersion Theory: Resonance Reactions 75 

section as well, and, if we introduce the same linear approximation as the 
expansion of (61), we have 

<r abs (0 = xX 2 • (21 + 1) _£=!• f68b) 

(E ~ E r )* + (r„ + r a ) 2 /2 {bm 



crj l) = 4ttX 2 -(21+ 1) 
in which we have set 



+ sin ae ta 



T n = -2kRP«)/~ 
dE 



(E - B r ) + (i/2)(r„ + r u ) 



r« = -2g/,_ 



«,(•> 



(68c) 



and defined the resonance energy E r by the relation 

(E r - E r «»)8 - A = (68d) 

E r = E r m + - with/ (#/°>) =0 5 = — 

S dE _ r 

Equations (68) are the principal results of the one-level theory, and they 
exhibit a number of interesting properties which we shall discuss briefly. 
2. Features of the One-Level Theory, (a) Level Shift. It will be ob- 
served that the resonance energy is increased by an amount A r = A/5 
from the value of the energy E r m at which the value of z was set equal 
to nir. Since this value of the energy is in any case not observable 
directly, nor calculable in the present theory, the shift might be regarded 
as meaningless. It has been pointed out, however, that in the com- 
parison of the successive levels of mirror nuclei, whose energy levels 
might be expected to differ by an easily calculated Coulomb energy, 
and in no other way, the effect of A r could be seen, for the excited states 
of two mirror nuclei will in general be capable of different modes of 
decay, since thresholds for charged particle emission will differ. Thus 
A r will differ for the two nuclei, and the observed resonances will not 
show energy displacement exactly equal to the Coulomb energy differ- 
ences. Qualitative agreement with the observations has indeed been 
found in at least one case (El), that of the pair N 13 , C 13 . Physically 
the level shift can be ascribed to the fact that in a compound level the 
particle which ultimately leaves the nucleus spends considerable time 
outside the nuclear radius and within the external force field before its 
departure; its wave function extends with sizable amplitude beyond the 
distance R. A more formal but somehow familiar description is to 
observe that such a shift represents the reactive part of that coupled 
impedance whose resistive part gives the familiar level broadening and 
damping. 



76 A Survey of Nuclear Reactions [Pt. VI 

(b) Reduced Widths. The particle width r„ is given by (68c) as just 

'-"""'(-ID" 

It is easy to compare this with the result of the compound nucleus ap- 
proach in Eq. (38). We see that here too we can define a "width without 
barrier," or reduced width, by writing F n = 2kRP ( - l) y r . The magnitude 
Y r is just the reduced width, which is dependent only upon the internal 
state of the nucleus, as described by the function /. Comparison with 
(38) shows that the reduced width is a measure of the probability of the 
particle being at the nuclear surface. The absorption width also can be 
written in this form, introducing a wave number k to represent some 
characteristic wave number for internal nuclear motion. We write the 
hitherto unspecified imaginary part of / near resonance in the form 
+g = icRha, and then we obtain r also in terms of the reduced width 
and of a dimensionless expression giving the imaginary part of /: 

r a = 2KRh a y r 

The observed width of a level is not always given by these formulas, since 
the variation of A in (68a) as the incident particle energy is varied is not 
always negligible; this effect can be computed by expanding A r itself 
about the resonance energy in (68a), and retaining only the linear 
variation. For s-wave neutrons, of course, A = 0, and there is no such 
effect; but cases have been exhibited in which the observed width differed 
from the value of r = V„ + r o by as great a factor as 2 or 3. 

(c) Negative Peaks. Formula (68b) for the scattering cross section 
exhibits some very odd properties, which arise from the possibility of 
interference between the resonance term and the second term, called 
the potential scattering term. This scattering term arises from the 
effect of the well-defined volume in which the nuclear forces act, and 
also from the summed influence of all the other levels of the compound 
system. There is no rigorous distinction between these two ideas; in 
our more formal picture we can think of the potential scattering as com- 
ing from the fact that between resonances the value of / is such that the 
surface wave must be of very small amplitude, much as though the 
nuclear surface were the surface of an impenetrable sphere. Taking 
the presence of a potential scattering term as our model gives it, then, 
let us examine its possible consequences, at least reasonably near a single 
resonance level. 

It is convenient to observe that the resonance term, with its varying 
denominator which gives rise to the familiar witch-shaped peaks, can 



Sec. 6B] The Dispersion Theory: Resonance Reactions 77 

be written in terms of a phase angle 0. With the notation 

%Tn r 

(E - E T ) + (i/2)(T n + r„) ~ A + i 

2(E - E r ) 



we write 



(r„ + r„) (r» + r„) 

r r 



= r sin Be 



A + i cotan + i 

which yields the very symmetrical form: 

crj l) = ttX 2 - (21 + 1)| r sin QeT m + sin ae ia | 2 

If we consider the resonance term alone, the familiar peak is produced 
by the variation with energy of the phase angle 0, going from a value 
of zero far below the resonance energy, taking a value of w/2 at exact 
resonance, and going to x far above the resonance energy. When the 
interfering term is considered as well, we may write the cross section 
in the following way to exhibit the interference : 

<7sc W 

* = — zJoi 7 = sm2 a + r2 sm2 6 + 2r sin 6 sin a cos (6 + a) 

Here it is clear that, if r = 0, we have pure potential scattering, which is 
constant with energy (for not too great changes in energy, we stay in 
the neighborhood of one single level) ; if a = 0, we have the pure resonant 
peak. The interference is described by the cross term, linear in r. A 
somewhat more understandable form of the expression can be obtained 
by a not very obvious transformation, using the trigonometric ex- 
pressions for sums and differences freely, and employing the iden- 
tity sin x + a sin (x + y) = (1 + a 2 + 2a cos y) M sin (x + e), tan e = 
a sin y/{\ + a cos y), which is best derived directly from the indicated 
geometrical construction. The transformed expression becomes 

r 2 r 
o- = h (1 — r) sin 2 a [r 2 + 4(1 — r) sin 2 a] 1A 



where 



+ r[r 2 + 4(1 - r) sin 2 a] y * sin 2 (0 + tf>) 

2 sin a cos a 

tan 24> = ^— (69) 

r — 2 sin a 



Here it is explicitly seen that the entire energy variation is contained in 
a sin 2 term, which recalls the general results of the method of partial 



78 A Survey of Nuclear Reactions [Pt. VI 

waves. The phase 6 of the resonant contribution always increases by ir 
as the energy rises through the resonance. But the value of the potential 
scattering phase angle <f> will determine the shape of the observed "peak," 
which will be a normal-appearing peak only if <f> = 0, x, and a negative 
peak when <j> = x/2, with a dip-and-peak combination for intermediate 
values. Such "negative resonances" have been observed, and the dip 
preceding a peak has been found in a number of cases. The scattering 
maximum will be displaced from the energy value E T of the absorption 
maximum because of the potential scattering interference. Such dis- 
placements are typically rather small, and no clear example can be 
cited. Use of (69) and related generalized forms to examine the proper- 
ties of particular levels, especially statistical weights and hence spins, is 
frequent. 

3. Spins and Statistical Weights. Up to this point we have ignored 
the existence of intrinsic spin for the fundamental particles involved in 
reaction, and of total angular momenta for the complex systems. We 
fix our attention on a single level of the compound nucleus. We shall 
assume that such a level has no degeneracies — all accidental ones being 
removed by coupling forces of some finite size, 'even if very small, 
within the nucleus — except the necessary degeneracy in spatial orienta- 
tion of the total angular momentum vector J. This implies, in the 
absence of external forces, a (2/ + l)-fold degeneracy of the compound 
state. But the compound nucleus can be formed in many ways. If the 
incoming particle has intrinsic angular momentum s, if we consider only 
the single orbital angular momentum partial wave I, and if the target 
nucleus has initial total angular momentum 7, then the total number of 
different ways to form a compound nucleus is (2s + 1)(2£ + 1)(2Z + 1). 
Of these only 2/ + 1 will correspond to the given compound level of 
angular momentum in question. Thus, for unpolarized beams incident 
on unpolarized target nuclei, and with no measurement of the spin of the 
resultant particle, we must multiply the cross-section formulas (68) by 
the statistical weight factor: 

2J+ 1 

° J ~ (2*+l)(2*+l)(2J+l) (70) 

The absorption cross section for the familiar one-level case with 1=0 
becomes, for example, 

2/.+ 1 . r„r a 

""abs — ~~~ 77ZZ T 7 ™ 



(2« + 1)(2I +1) (E- E r f + [(r„ + T a )/2] 2 

In general, of course, J is not known, and several possibilities exist. 
Even for the specially simple case of thermal neutrons, where only 



Sec. 6C] The Dispersion Theory: Resonance Reactions 79 

I = can contribute, and s = )4, we get the alternatives I + s = J + , 
I — 8 = J _, and the g J± factor is ambivalently J[l ± 1/(27 + 1)]. 
Sometimes J values can be assigned by study of the cross-section 
magnitudes. 

C. The Generalized Theory of Dispersion : Many Levels and Many 
Decay Modes 

We have treated the theory of reactions only in the simplest case. 
We have considered only two alternatives: the incoming wave is co- 
herently scattered, or it is absorbed. In general the theory must take 
account of many possible consequences of the formation of the com- 
pound state. It may be that the energy is not near a single resonance, 
but lies between two resonant values; or the widths may be comparable 
to the spacing, so that the effect of two resonances may overlap; or 
simply that compound states of different J can contribute to the emission 
of a single outgoing wave of fixed I, if spins are present. In all these 
cases, as in the case where several product particles are energetically 
allowed, we have to take into account the various possible courses for 
the reaction. 

In a series of papers (see Appendix II), Wigner and co-workers have 
presented a beautiful generalization of the process here applied in the 
one-level, two-alternative case. The phase shift rj which described the 
reaction by the relation <rj 0) = (x/A; 2 )(| 1 + v | 2 ) is generalized by 
introducing a unitary matrix U, such that 

^f = ~-\(U~l) if \ 2 

where i, f index initial and final particles, not only as to type, but also 
as to internal state (excitation), spin orientation, and relative orbital 
angular momentum. The theory now produces values for the matrix 
elements of (U - 1), often called the collision or S matrix, between all 
the states representing the various alternatives. Unlike the f(E) of 
our phenomenological theory, the matrices are given explicitly in terms 
of the Hamiltonian describing the interaction of all the nucleons in the 
total system. Progress is made, however, only by the demonstration 
that much of the behavior of the cross-section formulas can be studied 
by knowing precisely only the interactions outside the nuclear radius, 
and then replacing a detailed knowledge of the interior by certain bound- 
ary conditions on the nuclear surface. This is in strict parallel with the 
progress of the derivation we have given. Indeed, the more general 
method differs mainly by the complete generality into which it has been 



80 A Survey of Nuclear Reactions [Pt. VI 

cast, and by the somewhat more complete dynamical specification of 
the quantities which give the important results. The theory places the 
assumptions of our point of view in full sight, and demonstrates that 
the chief features of a theory of reactions come from the ability to specify 
with more or less definiteness a surface which can divide a region in 
which the Hamiltonian is fully known from one where it is not. We 
shall not further discuss the more general theory here, but refer instead 
to the literature. 

Here it is appropriate, however, to indicate one entirely formal scheme 
of generalizing our formulas, like (68), to include explicitly the properties 
of more than one level. It will be seen that there is full equivalence 
between representing the behavior of the cross sections between reso- 
nances (i.e., where more than one state must be taken into account) (1) 
as we have done it, following Weisskopf, by the use of a function f(E) 
whose properties, however, cannot be simply given except in the neigh- 
borhood of one level; or (2) by considering the summed contribution of 
a large, or strictly an infinite, number of levels, whose phase relationships 
and individual widths and locations can be known only in principle. In 
the absence of a detailed solution of the eigenvalue problem of the whole 
compound nucleus, no dispersion theory approach which is not statistical 
in nature can give useful results except in the neighborhood of a single 
level, or at most of a small number near-by, whose properties can be 
approximated. 

The formal equivalence of the two points of view comes out clearly by 
examining (66). We can write the denominator factor in this way: 

9(E) = r /T1N * - = £ 



fUE) - (A - n) j Sj - E 

This is an identity where the sum is to be carried out over all the poles «,- 
of the function g, i.e., the zeros of its denominator. Placing the sum 
into (62a) and remembering the linear expansion approximation, we 
can write the simpler formula of (62b) as 



Cabs — tX 



ZdY'iY)^/ 



E - Ej + \ (iv + IV) 



(71) 



The virtue of this way of writing the formula is that it exhibits the 
possible interference of the contributions of many levels. It is this 
interference which must be taken into account to explain angular dis- 
tribution of reaction products, especially in light nuclei, where broad 
levels are the rule. Here phase relations are evidently decisive. But 
the existence of constants of the motion, such as angular momentum 



Sec. 6D] The Dispersion Theory: Resonance Reactions 81 

and parity, will imply selection rules. The mode of formation of the 
compound nucleus can influence the phase of the matrix elements which 
occur in the (IV) H of the many-level formulas. In those cases the basic 
simplifying assumption of the independent decay of the compound 
nucleus will not be valid. 

When, as above, the total cross section, integrated over all angles of 
product emission, is calculated, any interference terms arising between 
compound levels which have different values of the constants of motion, 
total angular momentum, or parity drop out. However, for differential 
cross sections experimentally given by angular distribution or correlation 
measurements, these interferences are decisive. Indeed, every case in 
which the angular distribution of some product exhibits asymmetry 
with respect to a plane normal to the beam axis must arise out of such 
an interference between compound levels of differing parity. The 
best-studied example is that of the gamma-rays from proton capture in 
Li 7 near a strong 440-kev resonance. No general form of the com- 
plicated formulas involved is available, 1 though special applications 
occur quite completely worked out in the literature. The T r 's are best 
represented as matrix elements between the states involved, and close 
attention must be paid to the coherent I states in the incident beam, and 
to the various combinations of I, s, s z which can give rise to each com- 
pound level of fixed J r and parity. 

D. Statistical Estimates 

It is interesting to look at the value of o- abs (0) in the case contemplated 
by the statistical theory. We think of many levels contributing in a 
region of energy AE at E, with AE/E « 1 ; and we examine the average 
absorption cross section CT abs giving only average values for the properties 
of the individual levels. Formula (62) gives the contribution of each 
level. Replacing the quantity (E - E r ) by a variable of integration, 
and using the fact that the width V is small compared to the region AE 
so that we can treat X as constant over the level, we get, for each level, 



<W 0) &E = 



27r 2 x 2 r„r a 



(r n + r a ) 

and, for the entire set of N levels in the interval AE, 

. £ 2x 2 X 2 IYiy 2T 2 K 2 NT n T a 

(Tabs AtL = 2_, 



*■ n ~r *- a I n T la 

1 For an interesting account of the general properties of angular distributions see 

(Y3). 



82 A Survey of Nuclear Reactions [Pt. VI 

We then write for the number of levels its mean value, N = AE/D, 
where D is the mean level spacing, and we obtain 

... 2jt X r n T a _ 

<W 0) = — — — r = r a + r„ (72) 

It is clear that this expression is the same as the appropriate form of 
Eq. (40), with (49): 

(statistical) ^o ^ ^ .. .. o ^a „ 25rT n 



ffab. *•"■""*" = xX 8 •!).,». Pq« = tX 2 



r d 



where the sticking probability is now 2irY n /D, which is proportional to 
k ~ v at low energies. We have shown how the resonance contributions 
of the one-level dispersion theory sum to the statistical form if D >>> r, 
as we expect. 

Before we leave the dispersion theory, we shall cite one useful result 
which follows more directly from the generalized theory, or from the 
perturbation theory than from our phenomenological approach. The 
(reduced width) **, y r Vi , is shown in the generalized theory of Wigner (Jl) 
to be expressible as an integral over the nuclear surface of the product 
of two wave functions, one corresponding to the interior state of the 
nucleus, the other referring to the external motion of the product particle. 
The reduced width depends explicitly not only on the state of excitation 
of the compound nucleus, indexed by r, but also on the nature of the 
state of the residual nucleus, and therefore on the quantum state of the 
emitted particle. We have neglected the latter dependence, saying 
that only the external motion of the emitted particle was important. 
If, however, we sum over all the possible states / of the residual nu- 
cleus, we obtain a limit for y r which cannot be exceeded. This limit 
may be approximately evaluated by using the familiar sum rule for the 
product of the two matrices, y r / y2 , and regarding the internal wave 
function as expressing the fact that the nucleus has a roughly constant 
density. The upper limit thus approximated for y r is 

r 3 h 2 3 

Tr = i^P> - ~2^~I^ MeV ' f ° r nUCle ° n Width (73) 

This upper limit sometimes permits the exclusion of certain partial 
waves, when the low penetrabilities associated with high I would imply 
a y r violating the sum rule limit. The similarity between this limit and 
Eq. (39) is evident; from the general scheme it follows that we would 
expect widths near the limit for a one-body model, which would diminish 
as more and more levels contribute, or more and more particles take 



Sec. 7A] Some Typical Nuclear Reactions 83 

part in the nuclear motion. In general, our phenomenological theory 
would lead to the result that for a smooth enough variation of the func- 
tion / the value y r /D would be roughly constant (D is the level spacing). 
The departure from this result will measure the significance of devia- 
tions from the statistical theory. 

SECTION 7. SOME TYPICAL NUCLEAR REACTIONS 

In this part we shall apply the theory of the earlier sections to some 
typical nuclear reaction types which it has helped to explain. In each 
case a much more detailed account can be obtained in the original 
literature; it is the purpose of this section to indicate the method of 
using the theory in sufficient detail, and to point out as well its difficulties 
and pitfalls, so that the reader may extend the few examples given, 
necessarily briefly, here to the whole range of experimental material. 

A. Resonance : The Region of Dispersion Theory 

Clearly marked nuclear levels, and incident particle beams with well- 
defined energy, are known in only two types of reactions. The most 
important of these (discussed also in Part VII) is the interaction of slow 
neutrons with nuclei, leading usually to capture, but often to scattering 
or even particle emission. 1 The second type is the class of reactions 
using charged particle beams up to a few Mev of energy on target nuclei 
from the very lightest up to the region of, say, aluminum. In these 
reactions well-marked levels can often be found (see Figs. 10 and 11), 
the wavelengths involved are not small compared to nuclear dimensions, 
and the whole approach must be based on using the maximum informa- 
tion about individual levels. 

1. Thermal Neutron Reactions. This large class of reactions is dis- 
cussed in Part VII, Section 2B2. We shall here discuss it in sufficient 
detail to illustrate the use of the theory presented in Part VII. 

In the thermal region, the neutron wavelength is very large compared 
to nuclear dimensions. The neutron de Broglie wavelength X = h/mv 
is just X = 0.045 A/(E) V2 , where E is the kinetic energy in electron 
volts. For energies from a few millivolts up to, say, a thousand volts, 
which broadly defines the region of interest in these experiments, X 
ranges from 2 A to 10 -3 A. Throughout this range, the partial wave 
corresponding to I = 0, the so-called S wave, alone will be effective in 
reaching the nuclear surface. The penetrability of all other partial 
waves is so greatly reduced by the centrifugal barrier that they can be 

1 See, for example, the collection of results and references in (G6). 



84 A Survey of Nuclear Reactions [Pt. VI 

neglected. From this follows the isotropic distribution of scattered 
thermal neutrons and of capture gamma-rays in the center-of-mass 
system. 

(a) Effects of Target Motion and Binding. It is well to introduce here 
a note of caution. The usual neglect of the chemical forces acting on 
the target nucleus, of the thermal or zero^point motion of the target 
nucleus, and of the possible coherent scattering from the neighboring 
nuclei of the target material cannot be justified in the thermal neutron 
reactions. The coherent scattering of neighboring nuclei (discussed in 
Section 5 of Part VII) will show up strongly in cross-section measure- 
ments as long as the neutron wavelength is near the values which fulfill 
the Bragg condition rik — d sin 6 for the lattice spacing d of the crystal 
or microcrystals involved. This means that, in the range above 0.1 ev, 
the effect is not important for most target materials. The molecular 
binding effect is very important in determining the energy loss upon 
elastic collision, but again does not affect cross sections much in the 
region where the energy of the neutron is greater than the smallest 
vibrational level difference in the target molecule or crystal; and in 
addition this effect is of small importance for reasonably heavy nuclei. 
We shall treat here briefly the effect of the thermal motion of the target 
nucleus — the so-called Doppler effect. 

Let the velocity of the neutron be v in the laboratory system, and 
the component of the nuclear velocity toward the neutron beam be V. 
Then the relative velocity of neutron and nucleus is (v + V), and the 
relative kinetic energy 

fYh 

#rei = - (v + V) 2 = E n + (2mE n ) y *V (74) 

Jt 

to the first order in V/v. Here m is the neutron mass, E n the neutron 
kinetic energy. If the target atoms were those of a gas, they would 
move with the Maxwell distribution, giving for the fraction / of atoms 
having the velocity component V, with atomic mass A, 



"■ v > dv '(^rT e ~ m '" ir (75 > 



Inserting expression (74), we find, for the probability of a given E ie \, 
f{E iei ) dE tel = — • e-t*--™ "> — D = 2 [~~j-) (76) 
where D is the "Doppler width." The cross section is given by the one- 



Sec. 7A] Some Typical Nuclear Reactions 85 

level formula written in the form appropriate for neutron capture (or 
scattering), near the resonance energy, E r : 

Un 

<r(E„) = (77) 

1 + [(E n - E r )/(T/2)f K " } 

and E n here is of course to be replaced by the relative energy i? rel . 
The term {2mE n ) V2 V is the correction for the motion of the center of 
mass. With this we obtain, for the effective cross section S(E n ), 

5{E n ) = f*(E Tel )f(E Tel ) dE rel = c F (-' x) 

with x = (E n — E r )/(T/2), the deviation from resonance in units of 
the total width, and the integral 



\D / 2(tt)^Z>J_ 



e -rHx- X ') 2 /iD 2 ^ x '_ 



1+x' 2 

This function simplifies for large natural width to the form of (77) 
exactly as without temperature motion. But, for small natural width 
compared to Doppler width, the cross section at resonance is changed 
and the shape of the curve altered. In this case, with T/D <K 1, we 
obtain simple expressions for both : 

(1) E very near the resonance energy, (E n — E r )/D « D/T, when 

(*•)* r 



_ . 6 -(je„-fi.)7.D s 



2 D 

(2) E very far from resonance, (E n - E r )/D » D/T, and 

1 

^ 1 + [(E n - E r )/(T/2)] 2 

the value without Doppler motion. 

For E n = E r , exact resonance, we can easily obtain 

\D / 2 D \ (tt) 1a J ) 

If T/D 3> 1, this reduces to F = 1, and the cross section at resonance for 
large natural width is just a(E r ) = <r . For exact resonance, and small 
natural width T/D « 1, F(T/D, 0) = (r) K /2 • T/D, and the measured 
cross section at resonance becomes 

(»•)* r 



86 A Survey of Nuclear Reactions [Pt. VI 

much reduced by the Doppler broadenings. Lamb (LI) has shown that 
these same formulas hold even if the target — as is usually the case — is 
not a perfect gas but a Debye solid (other models will give similar 
results). In fact, if we replace D in (76) byanewvalueD = 2(£ , „-/cre q ) M , 
where T e(l is the equivalent temperature corresponding to the mean 
energy per vibrational degree of freedom (i.e., kT e<1 = mean vib. en- 
ergy/vib. degree of freedom), then the formulas are unchanged. This 
holds as long as either the natural or the Doppler width is large com- 
pared to the Debye temperature. For a not too sophisticated application 
of the Debye theory, the equivalent temperature is simply related to the 
Debye temperature. A few values are shown in Table 5. For other 

TABLE 5 

Equivalent Temperatubes or Cbystal Vibrations 



T eQ /T 


2 /"Debye 


1 


2 


1.06 


1 


1.15 


0.75 


1.35 


0.5 


1.8 


0.25 



cases, consult Lamb and the other references. It will be observed that 
all this discussion applies only in the case where the wavelength is such 
that neither crystalline nor molecular diffraction effects are appreciable. 
(6) Level Widths and Positions. With such conditions in mind, we 
can look into the Breit-Wigner one-level formula for neutron resonance 
absorption or scattering. From (62) or (70) we obtain 



a-X 2 / 1 \ a(E) 1A \T a forabs 

°abs,so = -— - I 1 ± — — — 7 ) — — TS : TTS X 



2 \ ± 2/ + 1/ {E - E r ) 2 + (r/2) 3 " [a(E) « for sc 

(78) 

Here we have written for the neutron width, r„, the value r„ = a(E) 1A , 
and we assume that the radiation width r a is constant. These two 
simplifications follow from the most marked feature of slow-neutron 
work: the fact that we are here studying, with resolution of a fraction 
of an electron volt, a very small portion of the level spectrum of the 
compound nucleus, some 5 to 8 Mev excited. We can safely regard all 
factors in the widths which depend on the compound nucleus as constant 
over the whole resonance, and from the formula we have the results 
employed. 

A few thermal neutron reactions are exothermic for heavy particle 
emission, as for example those with neutrons on H 3 , Li 6 , B 10 ; N 14 . Here 



Sec. 7A] 



Some Typical Nuclear Reactions 



87 



the widths are very large on the scale of thermal energies, many kilovolts 
at least, and the cross sections become simply <r a b s ~ X 2 (E) y * ~ 1/(E) lyi . 
This is the famous 1/v law. Even in heavier nuclei where gamma-emis- 
sion dominates, the 1/v behavior appears for a range in neutron energy 
small compared to the energy of the lowest-lying resonance. Only 
where the resonance lies very near zero energy (either above it as in 
Gd 157 or below it as in, say, Hg) does the 1/v law fail at low energies 
(apart from crystal and chemical binding effects, of course). 

A great body of experimental information has been obtained (and 
compiled in useful form by Adair (Al)) on the shape of the transmission 
curve and from it the cross sections for very many isotopes. These 
yield on analysis (for the narrow isolated levels for middle to high A) 
fairly reliable values for the three most characteristic parameters: the 
resonance energy E r ; the total width T and the neutron width evaluated 
at 1 ev energy; a, taken from the measured cross section at exact reso- 
nance, c(E r ). We tabulate a few typical values in Table 6. 

TABLE 6 

Some Slow-Neutron Resonances and Their Properties 



Nu- 


E r 


r 


aX 10 3 


CO 


Remarks 


Refer- 


cleus 


(ev) 


(ev) 


(ev)* 


(barns) 


ences 


Na 23 


3,000 


~170 


r« < r„/ioo 


550 


sc 


1 


Mn 55 


300 


~10 




~4,000 


sc 


2 


Co 69 


115 


2-5 




~12,000 


sc 


2, 3 


Rh 103 


1.21 ±0.02 


0.21 


0.5 


2,700 


abs 


4 


Cd 113 


0.176 ±0.001 


0.115 


2.2 


58,000 


abs 


5 


In 116 


1.44 ±0.03 


0.085 


2.2 


27,600 


abs 


6 


Sm 149 


0.096 


0.074 


2.0 


110,000 


abs 


7 


Eu 163 


0.54 


0.15 


1.7 


20,000 


abs 


8 


Gd 167 


0.028 


0.12 


4.5 


290.000 


abs 


9 


Au 197 


4.8 


<1 




>3,000 


abs (unre- 
solved) 


10 



1. C. T. Hibdon el al., Phys. Rev., 77, 730 (L), (1950). 

2. F. Seidl, Phys. Rev., 75, 1508 (1949). 

3. C. Hibdon and C. Muehlhause, Phys. Rev., 76, 100 (1949). 

4. R. Meijer, Phys. Rev., 75, 773 (1949). 

5. L. Rainwater el al,, Phys. Rev., 71, 65 (1947). 

6. B. McDaniel, Phys. Rev., 70, 832 (1946). 

7. W. J. Sturm, Phys. Rev., 71, 757 (1947). 

8. L. B. Borst et al., Phys. Rev., 70, 557 (1946). 

9. T. Brill and H. Lichtenberger, Phys. Rev., 72, 585 (1947). 
10. W. Havens el al, Phys. Rev., 75, 165 (1947). 



88 A Survey of Nuclear Reactions [Pt. VI 

It is seen that for all good absorbers the width r is indeed much 
greater than the neutron width, a(E) 1A . It is moreover reasonably 
constant among nuclei. This reflects the fact that it is made up of a 
sum of partial transition widths for the emission of gamma to all the 
possible lower states of the compound nucleus, which tends to average 
out fluctuations. But recent measurements on capture gamma-rays do 
appear to demonstrate a greater individuality among capture gamma- 
ray spectra than this point of view would lead one to expect; if those 
results are correct, we are seeing again a reflection of the special features 
of nuclear level spectra, perhaps arising from the importance of shell 
structure. 

A few examples of resonance scattering have been observed in some 
detail, both for slow neutrons, with I = 0, and for faster ones. The 
interference between potential and resonance scattering is plainly seen, 
along with the expected dip in the cross-section curves. Most thermal 
scattering, however, is simply scattering far from resonance — that 
contributed by the potential scattering term. The cross sections ob- 
served fluctuate more or less widely about the value 4x.ffi 2 , where R is 
the nuclear radius. The nuclear boundary cannot of course be very 
sharply defined for this process. Near-by levels will cause deviations 
which arise from the interference of other resonance terms. For the 
lighter elements, below A about 100, the level spacing is large, and the 
possible neutron widths therefore also larger (see Fig. 8) ; we may expect 
some sizable resonance scattering effects to show up, even rather far 
from resonance and without sign of much capture. The actual resonance 
scattering observed in Co and Mn is then complemented by some rather 
large deviations from the expected value, without evidence of actual 
resonance, in Cu, Ni, Fe, and a few other nuclei. Mn itself shows an 
abnormally low value of tr sc at thermal energy (H9, S6). 1 

The magnitude of the scattering cross section cr sc is measured of course 
by transmission. Recently experiments which make ingenious use of 
molecular and crystal coherent scattering effects have been applied to 
observe the phase change on scattering. These results demonstrated 
that most heavy nuclei showed a change in phase of w between incident 
and scattered wave (F4). This would be expected from the model we 
have used, for, wherever the "potential scattering" term was the most 
important, the phase change ought to be just that produced by an 
impenetrable sphere. The extension of the ideas of the one-body model 
made by the method of Weisskopf et al. still leaves unchanged many of 
the simple conclusions from that picture! 

1 This whole treatment follows closely that of (F7). 



Sec. 7A] Some Typical Nuclear Reactions 89 

The high distribution of high capture cross sections through the 
periodic table reflects the notion that the presence of a resonance level 
just at the excitation energy of the compound nucleus is a matter of 
sheer chance, since we have magnified a single half-volt region out of 
many millions. This consideration allows us to ascribe the fluctuations — 
which are indeed wild — to simple chance positioning of levels, and of the 
general trend to the gradual increase of level density. The cross sections 
for capture slowly rise, with fluctuations of course, until in the rare 
earths and beyond high cross sections are common. The sharp decline in 
the cross section of lead and bismuth is attributed to a decrease in 
neutron binding energy, also implied by the general kink in the mass- 
defect curve which is responsible for the natural radioactive elements. 
The level density, which ought to increase as the particle number in- 
creases for a given excitation energy, actually declines, partly because 
of the smaller excitation energy yielded by neutron capture, and partly 
because of the influence of closed shells. This is another reflection of 
the presence of structural detail superimposed on the statistical behavior 
of the nuclear drop; compare Fig. 17. 

2. Charged Particle Reactions on Light Nuclei, at Moderate Energy. 
The special ease of defining zero kinetic energy, which makes slow neu- 
trons so nearly monoenergetic, cannot apply to charged particle reactions. 
Slow protons are not very hard to make — hot atomic hydrogen — but 
they obviously will penetrate the Coulomb barrier so little as to make 
them useless for study of nuclear reactions until the temperature reaches 
that of stellar interiors, ~10 7 °K, where indeed thermal protons are of 
the utmost importance in inducing nuclear reactions, and are the 
agents of the release of stellar energy. But beams of charged particles 
up to a few Mev energy can be produced which are well-defined in 
energy and direction. Indeed the latest techniques * allow an over-all 
resolution, including the effects of the slowing of the protons in the 
material of the target, of the order of 100 ev out of a couple of Mev. 
For nuclei and excitation energies in a range where charged particles 
of this energy will penetrate the barrier, and where the level spacings 
are large compared to such a figure, we may expect to apply the dis- 
persion formalism to the computation of excitation functions. 

It will be clear that the general trend of all such reactions, which are 
primarily those with protons and alphas (or any charged projectile) 
on nuclei up to A ~ 50, will be governed by the effects of the Coulomb 
barrier. This will determine the width for entry T p and for re-emission 
of any charged particle. But, superimposed upon this easily understood 

1 See the work of the Wisconsin group as described in (H7), for example. 



90 A Survey of Nuclear Reactions [Pt. VI 

extranuclear factor will be the effects of strong resonances, which give 
the complex results seen in Figs. 10 and 11. We want to discuss here 
the method of treating such reactions; it is already obvious that only 
detailed attention to the properties of individual levels will be adequate, 
(a) The Reaction Li 7 + p. Figures 9 and 10 show the simplified 
experimental results for the two sets of products 2a and (7 + p')' — 
inelastic scattering — from the same reactants Li 7 + p. 1 The smooth 
rise in Fig. 9 is indeed just a penetrability curve, following the formula 

with a width without barrier, G, large compared to the energy variation 
for the protons, and an alpha-particle width which is independent of 
energy, since the reaction is exothermic by some 17 Mev and the alphas 
are far above the barrier. But the rise is not that appropriate for a pure 
Coulomb barrier, s-wave particles alone coming in without angular 
momentum; it is necessary to assume that the incoming particles pass 
as well through a centrifugal barrier, with 1=1. The »S-wave pene- 
trability would lead to a curve whose rise was nearly complete after 
about 500 kev; even more striking, the cross section observed is about 
fifty times smaller than such similar reactions as Li 6 (p,a)He 3 . This 
points again to the absence of s-wave particles, which experience a 
much smaller barrier. We must assume that such particles cannot 
induce the reaction from the compound state of Be 8 involved. The 
reaction is governed by a strong selection rule: the conservation of 
parity. The two identical alphas emitted obey Bose statistics; they 
must have a wave function invariant under their exchange. But to 
exchange two alphas is equivalent to reflecting their wave function in 
the origin, since they have no spin. The alpha wave function must 
then have even parity. Their relative orbital motion will have a parity 
of (—1)', where I is the orbital angular momentum. We have then 
shown that the parity of the system is even in the final state. Since 
parity is conserved, it must also have been even in the initial state. 
But all simple nuclear models agree in giving the ground state of Li 7 
odd parity. The incoming proton must therefore have had odd parity 
if the two alpha-particles are to be formed. This requires that the 
wave function for orbital motion of the incoming proton have an I such 
that $ ~ Yi m gives odd parity. The values I = 1, 3, ■ • • will do this; of 
these, the most easily penetrating partial wave is that with 1=1, the 
p-wave protons we observe. Thus we can account for the excitation 

'See (HI 1) and(H12). 



Sec. 7A] 



Some Typical Nuclear Reactions 



91 




o 



12 3 

Proton energy (Mev) 



Pig. 9. Relative cross section as a function of incident proton energy for the reaction 

Li\p,a)a. 



eg 

E 

C3 6 
bO 



S3 

o 

+3 . 
o 4 



^5 
13 




0.25 0.5 

Proton energy (Mev) 



0.75 



Fig. 10. Relative cross section as a function of incident proton energy, for the 

production of gamma-rays by the reaction Li 7 (p,p')Li 7 * in the region of the major 

sharp resonance at 0.44 Mev. 



92 A Survey of Nuclear Reactions [Pt. VI 

curve and the low yield by assuming even parity for the compound 
state involved. 

The angular momentum of the compound state must be the resultant 
of the incoming proton spin, S = J^i an d the target nuclear J = %, 
combined with the orbital angular momentum 1=1. This means that 
the total angular momentum of the compound nucleus Jc must be 
Jc = 0, 1, 2, or 3. But, if the state is to emit alphas, which have no 
spin, the angular momentum must be even (from the Bose statistics) 
and the state is then described by the quantum numbers J = or 2, 
even parity. The angular distribution of the alphas with respect to the 
fixed direction of the incoming beam can be no more complicated than 
the partial wave of the incoming beam, and in fact the wave function 
will be of the form if/ a ~ a + bYi°. Since Bose statistics requires that 
the wave function of the outgoing alphas be even in cos 6, the most 
general form possible is a + b(E) cos 2 6. This fits the experiment 
reasonably well up to some hundreds of kilovolts. Lately the higher- 
energy angular distribution has shown (H8) terms of the kind cos 4 6, 
which imply that incoming /-wave particles must be considered. The 
general features of the discussion are unchanged. 

But what of Fig. 10? The very sharp resonances for the emission of 
a 17-Mev gamma-ray must be explained. The cross section for this 
reaction at resonance energy, with I the spin of the target nucleus, is 

7rX 3 rj,r 7 27 + 1 

<Ty(E r ) ~ ■ 

(r p + r T ) 2 2 

and, since the measured value of <r y (E r ) is 7.2 millibarns, with a resonant 
energy of 440 kev and a measured total width of 12 kev (F9, F10), we 
can write the value of T y as 

ffyl abs 



r T ~ 



(2/+l),rX 2 -i 



assuming only that T y /V p « 1. This yields a gamma-ray width for 
the excited state of the compound nucleus Be 8 of about 30 ev. Ob- 
viously a strong selection rule must be in play to prevent the decay of 
the compound state by alpha-emission, which would be expected to 
have a width of hundreds of kev. The possibilities are two: (1) Only 
s-wave protons are captured to form this level, giving it angular mo- 
mentum Jc = 1 or 2, odd parity, or (2) the same p-wave protons which 
give the previous reaction are captured, giving states with even parity, 
but with spins J c = 1 or 3. In either case, whether the compound state 
in question has odd parity or odd J c , it cannot emit two alphas. To 



Sec. 7A] Some Typical Nuclear Reactions 93 

distinguish between the alternative, as to fix the J c in the two alpha- 
emitting states, requires further information. 

This has been provided by a beautiful experiment (D3) in which 
accurate measurements were made of the angular distribution of the 
gamma-rays. The experiments fully confirmed the dispersion formula 
in the interference form, Eq. (71), by the observed variation of the 
angular distribution as the proton energy was varied through resonance. 
Forward-to-backward asymmetry proves that two levels are involved in 
the gamma-emission, with opposite parities. It is natural to identify 
the wide interfering state with some state of the (p,a) reaction, neces- 
sarily even, and then the narrow resonant state must be odd, formed 
by s-wave capture. This is consistent with the observed isotropy of 
the gamma-rays near the sharp resonance. Two lower states take part 
in the gamma-emission: the ground state, transitions to which yield the 
famous 17.6-Mev gamma-ray; and a state at about 3 Mev excitation, 
nearly 2 Mev wide. Both these lowest states decay by alpha-emission. 
The general rule about the zero spin of even-even ground states would 
lead to the conclusion that the ground state here has J = 0. Some 
experimental evidence for a lifetime of this state near 10 — u sec does 
not seem strong enough to upset the rule, and evidence from the beta- 
decay of Li 8 (that the two lowest Be 8 states differ in J) supports the 
rule. The near-equality of the gamma-ray transition rates from the 
resonant state to the two low states suggests that they do not differ by 
many units in J. It seems likely that the 3-Mev state has J — 2, 
parity even. Then the choice between the two possibilities J e = 1 and 
J c = 2 for the sharp state seems direct; we assign the resonant state 
the quantum numbers J c = 1, odd, and all the effects are described; 
both gamma-rays are electric dipole, thus fitting with the sizable width. 

This neat account seems disproved by observations (C9) of the elastic 
scattering of protons at the 440-kev energy. The presence of a scatter- 
ing resonance whose interference with the Coulomb scattering has been 
studied shows that at least most of the compound nuclei are formed by 
p-wave, and not by s-wave, capture. Then the isotropy of the resonant 
gamma-rays is in the nature of an accident, (C4) implying not that the 
protons brought in no evidence of direction, but that the compound 
state with J = 1 was formed with equal probability in the three sub- 
states J z = ±1, 0. This would give isotropy; it will take place, accord- 
ing to the rules for combination of angular momentum matrices, if the 
triplet proton wave (spin parallel to orbital angular momentum) is cap- 
tured about one-fifth as frequently as the quintet wave (D4). Then 
the interfering state is some non-resonant odd state, and the sharp 
resonant state must be assigned the description J c = 1, even. The 



94 A Survey of Nuclear Reactions [Pt. VI 

gamma-rays are magnetic dipole and electric quadripole in a particular 
mixture. This is a less simple and less satisfying story, but it seems 
required by experiment. 

This account is a rather sketchy summary of one reaction, given as 
an example of the many-sidedness of the problem of nuclear reaction 
spectroscopy in the region where individual levels are of importance. 
The example at hand is not the simplest known case, but it is very far 
from the most complicated. The discussion of each reaction in great 
detail will be the content of a final nuclear spectroscopy. 

(6) Some Other Examples. Proton capture by C 12 and C 13 has been 
much studied. Here the fit to the one-level dispersion formula is excel- 
lent, though one must take into account the variations in T p coming 
from the appreciable change in penetrability over the 40-kev width of 
the resonance, which is some 2 Mev below the top of the barrier. It is 
worth while to note that the shape of the 440 kev resonance in the 
Li 7 (p,y) reaction does not fit the Breit-Wigner formula especially well. 
There is a superimposed asymmetry, with a rising gamma-ray yield 
on the high-energy side of the resonance. But in order to ascribe this 
to a variation in penetrability — arising from the capture of a proton 
partial wave with high I — a very high centrifugal barrier would have to 
be invoked (B19), and this would make the cross section absurdly small. 
Some more complex level scheme must be employed. 

The complicated and beautiful studies of the reactions arising from 
F 19 + p have led to repeated analysis from the point of view of the one- 
level formula (Fig. 11). Here the problem is to see if all the fifteen 
or more levels observed can be fitted into a scheme which gives reduced 
widths y r for the various reactions, each roughly independent of the 
level involved, and tries to account for the great experimental variation 
in width and strength of levels on the basis of penetrability changes due 
to* energy and angular momentum values alone. The situation is still 
not entirely unraveled, but the scheme is probably workable without 
too much arbitrariness (S2). 

There is increasing evidence of the value of another approximate 
constant of the motion, the so-called "isotopic spin" (W8). This 
quantity describes the spatial symmetry of the nuclear wave function 
under the interchange of any two nucleons. It can only be approxi- 
mately conserved, of course, since in fact the neutron and proton differ 
at least by Coulomb effects. Isotopic spin would be an accurate quan- 
tum number, and the nuclear wave function would consist of functions 
with one well-defined isotopic spin value, only if all nucleon-nucleon 
interactions were identical, independent of the type of nucleons involved. 
How strong this partial conservation law really is has not yet been estab- 



Sec. 7B] 



Some Typical Nuclear Reactions 



95 



lished. A similar situation exists for the spin, which can be intercon- 
verted with orbital angular momentum only in the presence of non- 
central forces. Strong forces of this type exist, and hence spin is in 
general not even approximately conserved; only total angular momen- 
tum, a strict quantum number, remains as a good basis for angular 
momentum selection rules. 

Whether the spectroscopy of nuclear levels based on the dispersion 
theory can be carried beyond these light nuclei is dubious. Even here 



A i ks^-J 




0.4 0.8 

Proton energy (Mev) 

Fig. 11. Relative gamma-ray yield (schematized) as a function of incident proton 

energy for the reactions coming from F 19 + p in the region below 1 Mev incident 

energy. Note the many and varied resonance peaks. 

the level schemes which have been proposed are highly complex. With- 
out an a 'priori guide to the position and character of nuclear levels, the 
job of spectrum analysis is difficult. It will be noticed that the problem 
of the light nuclear reactions is quite parallel to the problem of the low- 
lying states revealed in the gamma-decay schemes of radioactive nuclei. 
In the present example, however, the natural widths of levels is directly 
measured, while the number of possible reaction products adds the 
complicating factor of competition. But there seems little reason to 
doubt that the general scheme proposed is adequate. 



B. Reactions without Marked Resonance 

We have seen in Section 5 how the statistical theory of reactions 
proceeds by looking apart from the details of individual levels, and in 
Section 7A we have discussed in detail some reaction types which are 
clearly suited to the description of dispersion theory, in which the proper- 



96 A Survey of Nuclear Reactions [Pt. VI 

ties of single levels are made the basis for understanding. Here we shall 
apply the ideas of the statistical theory to the large class of reactions 
in which no marked properties of levels are observed; the theory is 
intrinsically suited to such reactions. We remark first that these reac- 
tions are generally characterized by high excitation of the compound 
nucleus. Wherever this occurs, the level spacing decreases and widths 
increase, as we have seen, so that either for all circumstances or for the 
conditions of nearly all practical experiments the importance of indi- 
vidual resonances is negligible. 

Excitation of nuclei by neutrons means the formation of a compound 
nucleus with energy of excitation equal to the neutron binding energy at 
least. As shown in Section 7A, this region (say 8 Mev in nuclei from 
A ~ 100 to about 200, and perhaps 6 Mev or less thereafter) is marked 
by discrete levels, spaced from a few hundred to a few electron volts, 
when the incoming neutron has negligible kinetic energy. As the 
kinetic energy of the neutron beam increases through the region of 
marked resonances, the level density increases rapidly. At a few kilo- 
volts at most, the present technique does not permit resolution of indi- 
vidual levels. We have already shown [e.g., in (72)] that the average 
over many levels of the dispersion formula leads to a result agreeing 
with the statistical theory. It is of interest to apply this idea to the 
actual case of the absorption of neutrons with energy in the range from 
a few to a few thousand electron volts, with the middle and heavy 
nuclei as targets. 

If we stick to s-wave neutrons only, we can write for the average 
absorption cross section, averaged over many levels, with level density p 



(0) = 2xp ■ xX 2 — — ; — — (bars indicate average value (79) 

over manv levels') 



J- n± a 

* ' a over many levels) 



[Compare (71).] From the expansion for f(E) given in (60) we can write 
r„ = — 2kR/8, with 8 and R constants which can be given a rough 
interpretation as determining properties of the function z{E) [see 
(62)ff.] or can be regarded as empirically determined. We find that 
r a y> F„ for energies below a few kilovolts, and then in this region we 
would expect 

-co, ^ R !^ 
" a ~ k ' s 

where R, p, a, 8 can be roughly estimated. Note that this applies to 
averages over many levels, not to thermal cross sections. Although 
there is no direct check yet on this prediction, it seems consistent with 



Sec. 7B] Some Typical Nuclear Reactions 97 

the general knowledge of absorption cross sections. A more detailed 
extension of this similar theory has given results 1 which are not wholly 
in accord with the data, at higher energies especially, but which seem 
to demonstrate the essential correctness of the approach, granting 
its rather high degree of arbitrariness in fitting the results of single 
experiments. 

The scattering cross section can be treated in the same way. Here 
of course the potential scattering term e* s sin S is important. Even for 
very simple assumptions about level widths, a rather complicated result 
is obtained which is not yet wholly confirmed. Both at thermal and at 
higher energies the importance of the "impenetrable sphere" effects of 
the potential term are evident in the total cross sections where no strong 
absorption resonance is present. Variations of radius R as well as special 
structural features of some nuclei seem to be present in the detailed 
comparison of theory with experiment. 1 Agreement is much better for 
some heavy and for some lighter nuclei; a few middle-weight nuclei do 
not give even the expected energy dependence of <r„ (0) , i.e., 1/E* A . 

1. The Statistical Approach. For sufficiently high energy of the bom- 
barding particle, no resonances will be observed. This can arise out of 
the experimental conditions : it is hard to define the beam to a very small 
energy range as its mean energy grows, and even the temperature motion 
of the target nuclei will produce a comparatively large uncertainty in 
relative energy for high bombardment energies, as we can see from 
relation (74). More significant, however, is the fact that the levels 
themselves become broader as the energy of excitation increases. This 
is the obvious consequence of the fact that higher excitation energy 
quite generally makes available many more modes of decay from a 
given level. In this region of high energy, then, the dispersion theory 
will have little value, for many levels, each with its unknown but specific 
properties, will take part in every reaction. 

In the preceding section we showed how the dispersion theory treat- 
ment went smoothly over into an entirely statistical form when the level 
spacing was small compared to the level width, for the particular case of 
neutron absorption. We replaced the precise values of the level widths 
by averages over many levels; such averages ought to vary smoothly 
with energy. In a sense we here calculate not the result of a particular 
reaction with a definite target nucleus, but a kind of average over many 
nuclear species very close in A and Z to the actual target. This ought 
to give then the general course of any reaction, fluctuations about the 
average behavior which will show up in particular cases being disre- 

1 ThisTvhole treatment follows closely that of (F7). 



98 A Survey of Nuclear Reactions [Pt. VI 

garded. The recent neutron scattering work of the Wisconsin group 
seems to show such fluctuations (B3). 

The fundamentals of the statistical theory were given in Section 4; 
here we propose to apply them. The main relations are two in number. 
The first, which is given in Eqs. (40) and (44), is the statement of the 
main Bohr idea of the nuclear reaction progressing in two independent 
steps: 

<r iiP = Sri ■ JS?- = xX* 2 E W + DPiiia ■ ■=£ = *i(fi) ■ ~~ (80) 

Here we have introduced the cross section <ri(c) for formation of the 
compound nucleus with incident particle i. The second is the statistical 
relation between level width for a given disintegration and cross section 
for the process inverse to the disintegration [see (33) and (34a)] : 

r ia (W) Pc (W) = ^ ~ (81a) 

and 

Y nl = kGP n ,i (81b) 

From these two we can find either the width or the cross section, using 
basic ideas, and compute the other from their relations. 

(a) Neutron Reactions. Here we have only the centrifugal barrier 
opposing the contact of neutron and target nucleus. Let us consider 
first the case of high-energy neutrons, with energies so high that the 
wavelengths corresponding are small compared to nuclear dimensions. 
In such a case we may follow the classical trajectory of the incoming 
particles. They will form the compound nucleus with sticking probabil- 
ity whenever they strike the nuclear disk. Moreover, it is evident from 
our whole nuclear model that for energies above, say, a few Mev the 
value of in is essentially unity: every particle that touches the nuclear 
matter sticks. Now we may write the contact cross section for neutrons 
in the partial wave of orbital angular momentum I as we did in Eq. (49) : 

tr n ,l = (21 + l)*K 2 Pn,itn.i (82) 

But for high energies we set £ n i -* 1, and the quantities P n ,i [found in 
(36)] take the values P n ,i — > 1 for low I's, such that the centrifugal 
barrier is well below the energy available, say for I below l c , the critical 
angular momentum. For I > l c , the values P„,; rapidly approach zero, 
and, if many Z's are involved, we can neglect the transitional cases near l c . 



Sec. 7B] Some Typical Nuclear Reactions 99 

Then just as in (44b) the cross section <r n (c) for formation of the com- 
pound can be written 

C n (c) = 2(TiZ 

l c ~hR 

= ttX 2 23 (2* + l)P,m -» ttX 2 E (2Z + l)P Bl! 
2=0 o 

^irP 2 (high energy) (83a) 

which is of course the classical value. 

At low energies, we can give no a 'priori guess about the quantity 
£„,i. But relations (81) for level width can be applied. We write 

Tm = kyP ntl = ^- . f=I = i*- (2Z + l)P n fa (84) 

2:rp irX 2irp 

It now seems inviting to make the identification : 

2irp7 

£„,* = * (85) 

gn 

Here the functions p and 7 depend only on the compound nucleus 
[while gR is the statistical factor needed in <r n i, by (69)], and will vary 
very little for small changes in neutron kinetic energy. We write this: 

k 
£» = - 

and 

k 
<r n (c) = ttX 2 2(2Z + l)P ni! • - (low energy) . (83b) 

K. 

with the constant K to include the unspecified functions above. The 
result is then a sticking probability which is proportional to k for slow 
neutrons, which can be justified from the side of the dispersion-theoretic 
discussion in Section 5. As energy increases, the sticking probability 
goes over to 1 gradually, and we may choose the constant K to give a 
smooth extrapolation from low- to high-energy values, from (83a) to 
(83b) . The resulting neutron cross section behaves like 

c n = icX 2 -P ,i-{k/K)~l/v 

at the lowest energies, where only the s wave, with I = 0, contributes 
and gradually falls with increasing energy until it approaches the geo- 
metric cross section. 

It is to be remembered that all this refers to the cross section for actual 
formation of a compound nucleus. The "potential scattering" term 
e id sin 5 of (68) is present in addition. Part of the compound nucleus 



100 A Survey of Nuclear Reactions [Pt. VI 

formation cross section may indeed lead to re-emission of a neutron 
with just the incident energy, a process of elastic scattering after com- 
pound nucleus formation. 1 This part can in fact give coherent inter- 
fering contributions to the diffraction effects. Such interference terms 
can complicate the shadow scattering. In principle we would expect 
angular distributions of elastically scattered neutrons to show a more or 
less isotropic part, coming from the re-emission of absorbed neutrons 
from several overlapping levels in the compound nucleus, together with 
a more complex and usually well-collimated part, which would contain 
the diffracted shadow waves plus interference terms from both sources. 
In the statistical theory, however, we expect the part of the cross section 
of compound formation we discuss to approach the value wR 2 for high 
energies, though the total cross section, including the elastic diffraction 
or shadow scattering, will become very nearly twice that. 

(b) CJiarged Particle Reactions. Here the Coulomb barrier introduces 
an added complication. We begin with the last member of (80) : 

<n(e) = 7rXi 2 2(2Z + 1)P„{« (80) 

It is now in the spirit of our statistical approach to give the sticking 
probability the same form for charged particles, once they have reached 
the nuclear surface, as we found it to have for neutrons, for the nuclear 
forces are predominant there. If now we write the W.K.B. expression 
for the penetrability P v j from Eq. (37), we have, in the region E/B « 1, 
where the W.K.B. method is reliable: 

<r,-(c) = ttX 2 2(2Z + l)fc, • (^) V 2c < = ,rX 2 2(2Z + l) e " 2ci T^?~ 

Now the constant K can be chosen to make the cross section for the low- 
energy, high-barrier region, where the W.K.B. expression is adequate, 
go smoothly over to the high-energy limit, which is just 

aiic) = ttX 2 2(2Z + l)P n ,i -» x£ 2 

with the same arguments about the critical angular momentum l c as 
in the case of the neutron. We finally obtain then, for incident charged 
particles, i, 

d{c) = xX 2 £ (21 + l)e- 2Cl for all energies (86) 

1=0 

A long literature exists on the points we have treated here quite heuris- 
tically. By and large, the simplest justification for our treatment is 
found in the related discussion of Section 5, but compare also the papers 

1 Cf. discussion following Eq. (45). 



Sec. 7B] Some Typical Nuclear Reactions 101 

of Konopinski and Bethe (K8) and Bethe (B13), whose results we have 
essentially set out above, but by rather different methods. 

We include here a number of graphs (Figs. 12 to 14) showing the re- 
sults of computations of the cross section for compound nucleus forma- 
tion, based on (83) and (86), and using the same value both for the target 
radius {R — I A- A ^-10 -13 cm) and for the radius of the projectile. 




X -^proton 

Fig. 12a. Cross sections for protons, for formation of the compound nucleus, plotted 

linearly. 

The graphs apply to protons, alphas, and neutrons over a wide range 
of energies, for typical target elements over the upper two-thirds of 
the periodic table. Following these graphs, we include graphs (Fig. 
15) of the specific widths, the functions defined in (50b). These graphs 
are based on the cross sections given in (83) and (86), and on the as- 
sumption concerning the statistical level density of compound nuclei, 
given in (31). From the specific width graphs we may compute the 
function D p = I^/STy = /p/S/y for any definite reaction, and hence 
the total reaction cross section on the statistical theory. The special 
problems raised by the deuteron and the gamma-ray will be discussed 
later. The specific widths are given for typical nuclei as functions of the 
energy of excitation of the compound nucleus, for the several possible 
emitted particles. 



102 



A Survey of Nuclear Reactions 



[Pt. VI 







0.2 



0.4 0.6 0.8 

X = "proton /^barrier 



1.0 



Fig. 12b. Logarithmic plot of proton cross sections. 




Fig. 12c. The proton barrier height needed for the cross-section graphs, as a function 

of Z. The parameter x is given in terms of proton kinetic energy measured in the 

laboratory frame. Several values for R have been assumed. In the cross-section 

curves we use R = 1.4(A M + 1)10 -13 cm. 




x-E a /E barrier 



Fig. 13a. Cross sections for alpha-particles, for formation of the compound nucleus, 

plotted linearly. 
103 



104 



A Survey of Nuclear Reactions 



[Pt. VI 



a! 




0.5 0.6 0.7 0.8 



0.9 1.0 

Fig. 13b. Logarithmic plot of alpha cross sections, for energies below the barrier. 



x -E^/E h!Lrrier 



15 



> 



.SP 10 






5 - 




10 



20 



30 



40 



50 
Z 



60 



70 



80 



90 



100 



Fig. 13c. The alpha-barrier height needed for the cross-section graphs, as a function 
of Z. The parameter x is given in terms of alpha-particle kinetic energy measured 
in the laboratory frame. (Lowest barrier has been used in cross-section graphs.) 











10 


\\ ° 






£ 5 

b 


1 


-— — ° 


o experimental points for Ag 
£=90 O 




Z=GQ 




1 1 


Z=30 





i i i i i 



123456789 
tf„(Mev) 

Fig. 14. Cross sections for neutron absorption in three elements. 

105 



10 



106 



A Survey of Nuclear Reactions 



[Pt. VI 



The same calculations which lead to the specific widths give (before 
integration) the energy spectrum of emitted particles. We have already 
shown a typical set of spectra in Fig. 6. 




11 



13 



15 



9 
R (Mev) 

Specific widths for neutron, proton, alpha-particle, and gamma-ray emis- 



7 
E, 



Fig. 15. 

sion from several compound nuclei, as a function of energy of the emitted particle. 
The specific widths are computed for residual nuclei of even A and Z. For odd-even 
or even-odd residual nuclei, multiply the plotted width by 2; for odd-odd residual 
nuclei, multiply plotted width by 4. Interpolation for other Z can be made directly. 

The application of these formulas to all the reaction types listed, in 
the appropriate energy region, is straightforward. To point out a few of 
the more general consequences, and to provide a model for calculation, 



Sec. 7B] Some Typical Nuclear Reactions 107 

we shall discuss one or two particular reactions which can be compared 
with experiment. 

(c) Competition. The reactions (a,ri) and (a,2n) on Ag 109 have been 
studied by Bradt and Tendam (B20, Gl). Figure 16 is a simplified 
version of their experimental results, and of some later work on similar 

300 

O In(oi, n) 

7 Ag i? = 1.4-(A 1/s + 4Vs). NT 18 cm 

Ag(a,«)-A ; 
§200 



0) 

> 
§100 




Sn i? = 1.5-^ 1/a -10- 13 cm 



14 15 16 
E (Mev) 

Fig. 16. Comparison of experimental and theoretical cross sections. The circles 
refer to the measured relative cross section for In(<*,n) (see reference Gl), and the 
triangles to the work of (B20) for Ag(a,n). Two theoretical curves are fitted: one 
with R = 1.5-A^-lO- 13 cm, and one with R = 1.4 -(A* + 4«)-10- 13 cm. The 
difference is a fair indication of the limit of reliability of our statistical theory as 
well as of the assumption R ~ A^. 

reactions. The cross section for formation of the compound nucleus, 
with sticking probability near unity for these energies, is plotted as well 
(the graphs of Figs. 12 to 14 were used). The compound nucleus can 
emit a neutron, two neutrons, charged particles, or gamma-rays. The 
nucleus is excited by the energy E a + B a . The binding energy of the 
alpha, B a , is not known, but can be written B a = — 4B„ + 28 Mev, 
where B n is the average binding energy of each of the nucleons brought 
in, and the mass defect of the alpha is known to be 28 Mev. We cannot 
give B n with any accuracy in this region of the table, but it is pretty 
surely between, say, 4 and 8 Mev. The compound nucleus is then 



108 A Survey of Nuclear Reactions [Pt. VI 

excited to between 15 and 20 Mev in these experiments. Such an excita- 
tion energy brings us well into the region of dense levels. The escape 
of a gamma-ray will be negligible here, for, as we shall see, the gamma 
widths are always small compared to heavy-particle widths when the 
excitation is appreciable. 

Charged particle widths will also be small compared to neutron widths, 
even though the excitation energy is a good deal beyond the barrier for 
protons, because, from expression (52) the release of a proton with the 
full available energy is not probable. It is probable that the emitted 
particles take off energy corresponding to the nuclear temperature, here 
some 2 Mev, and for such protons the barrier is still effective. We 
expect that neutrons will go off almost all the time: T n ^> T p . The 
results of experiment agree nicely with this idea from the lowest energies 
measured up to about 15.5 Mev. Here the rate of increase of the cross 
section for (a,n) drops sharply. With alpha-particle energy of 4 Mev 
higher, the value of tr a , n has fallen to a third or less of the value of 
<r a (c). This is the typical effect of competition. The factor D n has 
changed from very near unity to something much smaller because the 
threshold for a new reaction has been reached at 15.5 Mev. The new 
reaction is the (a,2n) reaction, which leaves the residual excited nucleus 
In 111 . We may estimate the probability of this reaction very roughly 
as follows: The cross section for formation of the compound nucleus is 
<r a (c), given in Fig. 13; £ is about unity; and the fractional number of 
neutrons emitted with energy E n is, from (52), just 

— — = E n <r n e E " /T (87) 

In order to emit a second neutron, the first neutron must leave the 
residual nucleus sufficiently excited to emit the second one. This will 
occur whenever the residual nucleus has even slightly more energy (by 
a few kev) than the binding energy B n of the neutron in the residual 
nucleus, for the gamma-width is then negligible, and charged particle 
widths are all extremely small because of the Coulomb barrier. We 
can then write 

In(En) dE n 
a(a,2n) = a a (88) 

" En, max 









I n (E n ) dE n 



where the integral in the numerator is carried from zero removed energy 
to AE = E n max — r 2n , with V 2 „ the threshold energy for emission of a 
second neutron. Both integrals can be approximated by using the 



Sec. 7B] Some Typical Nuclear Reactions 109 

Maxwell distribution of (87) : 

1 - (1 + AE/ T )e- AE/T 
<r{a,2n) = <r a (89) 



1 



(i + ^=). 



max/T 



where AE = i? max — r 2 „ is the excitation energy surplus beyond the 
threshold of the 2n reaction, i? max = E a + B a — B n . The threshold 
for the observed reaction in silver is 15.5 Mev, and E m&x ~ 2 or ZAE, so 



21- 



OJ 1 

£1 



Bi 209 

i2 = 1.4-(A 1/3 + 4 1/3 )-10" 13 



Ni 60 



■ a 

m 



i? = 1.3-U 1 /3 + 4 1 /3).i -ii^^ / ,. (a,2n) 




(a,w)+(a,2w) 



□ (<*.n) 



10 20 30 40 

E a (Mev) 

Fig. 17. Comparison of theoretical contact cross sections with the sum of observed 

(a,n) and (a,2n) cross sections. The solid curves are the theoretical values and are 

marked with the radii assumed. Ni 60 data from (G2); Bi 209 data from (K5). 

we can set the denominator equal to 1, and the fit with theory is then 
excellent up to 18 or 19 Mev, as far as the experiments go. Agreement 
was obtained by choosing the best value of the temperature r, which 
turned out to be the very reasonable value 1.8 Mev. Our simple model 
(Table 4) gives this value for silver excited about 18 Mev, which is not 
far from what we would expect for the excitation of the residual nucleus 
to the limit of the neutron spectrum, with alphas of about 15 Mev. 
Other multiple reactions, as (n,2n), and even the more highly multiple 
reactions, may be treated in a similar fashion. The literature contains 
discussions of the most characteristic features of many other reaction 
types (see Appendix II). We include more experimental comparisons 
in Fig. 17; the entry of successive competing reactions is beautifully 
shown (G2, K5). 



110 A Survey of Nuclear Reactions [Pt. VI 

SECTION 8. THE DEUTERON AS A PROJECTILE 

We have not so far discussed what is the most commonly used nuclear 
projectile excluding pile neutrons. This is the deuteron, 1H 2 . It is 
essentially different from the proton and the alpha-particle, because 
neither is it a simple nucleon, nor is it so tightly bound, like the alpha, 
that its state of lowest internal energy alone plays a part in nuclear 
reactions at ordinary energies. On the contrary, the deuteron is so 
loosely bound — only (2.23)/2 Mev per nucleon — and so large a structure, 
with its constituent nucleons often 3 or 4 • 10 -13 cm apart, that these 
properties play the major part in determining the course of deuter on- 
induced reactions. 

The use of deuterons to produce nuclear reactions, especially for the 
copious production of radio isotopes, is by now traditional. An excellent 
set of data (C7) has been published which gives the practical yields — 
often quite large — from deuteron bombardment at moderate energy. 
We give a few typical numbers in Table 7. Perhaps the most evident 





TABLE 7 




elected Experimental Yields with 


14-Mev Detjt 




(Thick-Tabget) 


Yield 




Product 


(nuclei/10 6 


Reaction 


Isotope 


deuterons) 


Mg 24 (d;a) 


nNa 22 


500 


Na 23 (d;p) 


nNa 24 


1400 


AF(d;p,«) 


u Na 24 


5.7 


p 3I W;p) 


15P 32 


860 


Cr 52 (d;2n) 


2S Mn S! 


140 


Cu 63 (d;p) 


29CU 64 


270 


Br 81 (d;p) 


36 Br 82 


220 


Te 130 (d;2n) 


63l 130 


95 


Te 130 (d;n) 


63 I 131 


180 



consequence of deuteron bombardment is the high excitation of the 
compound nucleus formed by deuteron capture when it takes place. 
The nucleus is excited by the kinetic energy of the beam plus the full 
binding energy of a proton and a neutron, reduced only by the weak 
2-Mev binding of the free deuteron. This amounts in general to some 
14 Mev plus the kinetic energy. Such an excitation implies the validity 
of the statistical theory. The penetrability may be calculated as in 
(37) and in Figs. 12 to 14 in a fair approximation to the deuteron contact 
cross section, for a deuteron of energy Ed on a nucleus of charge Z is 



Sec. 8A] The Deuteron as a Projectile 111 

that for an alpha-particle with energy (2^E d ), but with a target nucleus 
of charge Z/2 Vi , i.e., an alpha with E' = \AE d , Z' = 0.6Z. (This 
implies the rather rough approximation, A proportional to Z.) 

Such deuteron-induced reactions would lead nearly always to neutron 
emission, and often to multiple-particle emission. The neutron can 
take off sizable energies because of the high nuclear temperatures. It 
is nevertheless striking that many reactions have been observed, espe- 
cially with the heavier nuclei, in which proton emission was of com- 
parable frequency with neutron emission. This is the case not only at 
higher energies, where the proton barrier might be unimportant for the 
nuclear temperature involved, but even at moderate energies, and espe- 
cially in the heaviest nuclei. We should like to discuss this phenomenon 
in detail; it is the result of break-up of the deuteron, and it is called 
stripping. 

A. The Oppenheimer-Phillips Reaction : Low-Energy Stripping 

The deuteron moves toward the nucleus in the presence of the large 
electrostatic field produced by the nuclear Coulomb charge. The proton 
in the deuteron is repelled from the nuclear surface, but the neutron feels 
no such force. For the highest energies, or for light nuclei, when the 
Coulomb barrier is unimportant, the deuteron is polarized by the Cou- 
lomb field, but it may not break up. In cases where the Coulomb barrier 
is important — say comparable to the binding energy / of the deuteron — 
the proton will not reach the nuclear surface when the neutron does. 
The Coulomb forces will repel the proton, and the nuclear forces will 
seize the neutron. The compound nucleus then will be formed not by 
the capture of a deuteron, but only by that of a neutron, and the proton, 
which will never have come within the range of the nuclear forces at 
all, will fly off with a gain in kinetic energy, both from the recoil of the 
broken deuteron bond, and from the Coulomb repulsion. The ratio 
of the barrier height to the deuteron binding energy will be a very rough 
measure of the importance of the process. This process is sometimes 
called, after the first authors to recognize it, the Oppenheimer-Phillips 
process (01). 

The process can be treated by dividing it as usual into several steps. 
We write the cross section for a reaction initiated by a deuteron of 
energy E d resulting in the release of a proton of energy E p in the interval 
dE p as follows : 

a(E d ,E p ) dE p = <r d (po,R)lT(E p ) dE p (90) 

Here a d is the contact cross section for penetration of the proton com- 
ponent of the deuteron, not to the nuclear surface, but to the more or 



112 A Survey of Nuclear Reactions [Pt. VI 

less well-defined distance p to which the proton has the maximum 
probability to penetrate; T(E P ), called the transfer factor, takes into 
account the gain in energy of the proton as it moves out, and £ is the 
sticking probability of the neuteron averaged over all available levels 
into which it may be captured. 

The modified penetrability function for the deuteron, ad(p R), taking 
into account the polarization produced by the Coulomb field, has been 
given, with useful graphs (VI). The factor £ will increase with energy 
of excitation, perhaps as rapidly as the level density, and will become 
unity for high excitation energy, which means low proton energy. High 
proton energy will mean low excitation energy, and may indicate the 
existence of individual levels. For low deuteron energies, the excitation 
of the nucleus will be quite low, even less than that following slow 
neutron capture, since the binding energy of the deuteron will be lost to 
the compound nucleus. The neutron sticking probability can be quite 
low at such "negative kinetic energy" but, even then, the easy escape 
of the proton from its rather distant point of closest approach will result 
in a large Oppenheimer-Phillips cross section and a small deuteron 
capture. This cross section will be larger than deuteron capture just 
for this reason, even when the neutron does not always stick. 

For higher deuteron energies, the proton can approach the nuclear 
surface rather closely, and the most probable distance, p , will not be 
much greater than R. Then the ordinary deuteron capture and neutron 
emission will have an excitation function much like that of the Oppen- 
heimer-Phillips (d,p) process. But, by reason of the high excitation 
following capture of a deuteron, the (d,p) process dominates the (d,n) 
process up to energies well above the barrier, since the (d,pri) process 
is very likely to occur after deuteron capture. 

B. Stripping Reactions at Higher Energy 

Even at very high energies, of course, the passing deuteron is dis- 
torted by the nuclear Coulomb field, and the Oppenheimer-Phillips 
process takes place. Here one may think not so much of the purely 
electrostatic forces as of the passing deuteron in a varying field due to 
the nucleus. In a frame of reference in which the deuteron is at rest, 
and the heavy nucleus is moving almost undeflected past it, the electric 
field experienced by the deuteron is a highly transverse field, owing to 
the Lorentz contraction of the nuclear force lines. This pulse of electric 
field can be Fourier-analyzed into a collection of nearly transverse 
waves of a wide spread of frequencies. These waves behave nearly 
like quanta, and are often called "virtual quanta" in this method of 
calculation (Dl). The virtual quanta may induce photodisintegration 



Sec. 8B] The Deuteron as a Projectile 113 

of the deuteron as they pass by. In the laboratory system, where the 
deuteron is moving, the process becomes an electromagnetically induced 
stripping. It is in general a minor contributor to deuteron reactions 
compared to specifically nuclear effects at high energy. 

But, much more important in the high-energy limit, where the deu- 
teron may be handled classically, with a well-defined trajectory, is a 
process we may call collision stripping (S7). Here the deuteron is 
broken up by actual contact of one of its particles with the nuclear 
matter, while the other particle flies off with the momentum it had at 
the instant of collision, the resultant of its share of the deuteron center- 
of-mass motion and of the internal motion within the deuteron. This 
process yields neutrons and protons in equal numbers, and in a cone 
restricted to the forward direction, with a half-angle given by the simple 
relation 6 ~ (W d /E d ) V2 , and energies given by E p ~ E n ~\(E d V2 
± Wd Vi ) 2 , where W d is the deuteron binding energy. The process is 
evidently one which leads to different energy and angle distribution from 
the low-energy Oppenheimer-Phillips reaction, or from emission after 
formation of a compound nucleus. Although detailed agreement with 
this simplified form of the theory is to be expected — and has been veri- 
fied — only for really high-energy deuterons, at 190 Mev, the process 
must contribute continuously at lower energy, and finally merge with 
the standard Oppenheimer-Phillips case, in which the "stripping" 
occurs with by no means negligible reaction between the captured 
nucleon and the one that goes free, and in which the trajectory is spread 
heavily by diffraction (P2). 

With deuterons in the region of moderate energy, say from 5 to 20 
Mev, the stripping process is still dominant. Forward-peaked and even 
more complicated angular distributions of neutrons from these projec- 
tiles on a variety of targets of nearly any range of A indicate that neither 
compound nucleus formation nor a statistically treated stripping, in 
which classical ideas are used and all orbital angular momenta are 
regarded as contributing, can completely account for the process (B23). 
With good energy definition of the incident beam, it is possible to fix 
upon a group of outgoing protons of a definite energy. These protons 
arise from the capture of the neutron into one specific level of the prod- 
uct nucleus. At these moderate energies the excitation of the nucleus 
may be less even than that following capture of a free neutron, because 
the deuteron binding energy must be supplied to free the outgoing pro- 
ton. This implies that individual levels will separately contribute. 
Since the incident deuteron momentum in the stripping approximation 
point of view is just equal to the sum of the momentum of outgoing 
proton and of the neutron before capture, forward-emitted protons may 



114 A Survey of Nuclear Reactions [Pt. VI 

carry most of the available momentum. The internal deuteron motion 
does not often permit large values of momentum for both nucleons. 
Thus the captured neutrons frequently have long wavelengths, and 
quantum effects become decisive. The loose deuteron structure means, 
moreover, that large values of the orbital angular momentum of neu- 
tron with respect to nucleus can be important, while only a few I values 
can be captured into a single level. Then penetration effects may show 
up strongly in the angular distribution of the stripped protons, espe- 
cially at small angles. Study of the fine structure of these distributions 
determines the partial waves captured, and becomes a powerful means 
of level spectroscopy (B22). 

The really complicated behavior of the deuteron as a projectile, which 
arises from its loose structure and internal motion, seems qualitatively 
explained by the various processes outlined; it is by no means clear in 
full detail. 



SECTION 9. RADIATIVE PROCESSES IN NUCLEAR 
REACTIONS 

The all-important role which radiation plays in the de-excitation of 
atomic states is diminished in nuclear reactions by the numerous alter- 
native means of decay. In nearly all reactions, nevertheless, gamma- 
rays are observable products, since particle emission cannot always be 
expected to leave the nucleus in the ground state, and, once the cooling 
nucleus has dropped below the energy content representing the threshold 
for particle emission, only radiation can take away the remaining energy. 
With the new machine sources of continuous gamma-spectra, moreover, 
nuclear reactions induced by gamma-rays are of high interest. The 
fact that the electromagnetic interaction between field and charge- 
current is completely known makes the study of nuclear structure 
through radiation, rather than particle collision, seem attractive. The 
somewhat illusory nature of this argument comes both from the ex- 
perimental difficulty of accurate measurement for processes of generally 
low yield under conditions of poor energy resolution and from the com- 
plex character of the charge-current vector within nuclei, which turns 
out to depend on rather fine detail of nucleon motion, as on the presence 
of exchange forces and other phenomena connected with the intranuclear 
motion of the meson cloud. But the whole subject is full of interest. 

A. The Multipole Classification 

Just as the emitted particles are classified by the orbital angular 
momentum of their partial waves, which determines angular correla- 



Sec. 9A] Radiative Processes in Nuclear Reactions 



115 



tions and penetrabilities, so it is useful to classify radiation from any 
charge-current system contained in a limited volume of radius R in a 
similar way, according to the successive terms of a general expansion. 
This expansion is called the multipole representation, and it amounts 
to a sorting by angular momentum and parity. The electromagnetic 
field is a vector field, with some special properties due to the zero mass 
of the particles of the field — photons — and it turns out that there are 
two independent partial waves (analogous to the familiar two types of 
polarization) for each value of total angular momentum radiated away. 
Speaking physically, one would expect three possible angular momentum 
values for each orbital value, since the photon has unit spin. But the 
condition of transversality, which is closely related to the zero mass 
of the photon, excludes one orientation. In Table 8 are presented the 







TABLE 8 
Multipole for Given Transition 




Parity 


Radiated Angular Momentum, J y 


Change 





1 


2 


3 


L (even) 


L (odd) 


Yes 
No 


None 
None 


Elec. dipole 
Mag. dipole 


Mag. quad- 
ripole 

Elec. quad- 
ripole 


Elec. octo- 

pole 
Mag. octo- 

pole 


Mag. 

2 L -pole 
Elec. 

2^016 


Elec. 
2 i -pole 

Mag. 

2^016 



selection rules and the type and order of multipole for a given parity 
change and radiated angular momentum. The radiated angular mo- 
mentum is of course to be taken in the usual sense of the vector model : 
if Ji and J/ are initial and final angular momenta, the relation J t - + J y 
> Jf > | Ji — J y | must be fulfilled. 

In general only the lowest order of multipole allowed by the selection 
rules contributes; e.g., if the transition is Ji = 2, odd —►«// = 1, even, 
we expect electric dipole only. The magnetic multipole of order L is 
reduced in intensity with respect to the electric dipole of the same order 
in the ratio (h/McR) 2 . For electric and magnetic dipole this is easy to 
see by simply writing the ratio of the familiar dipole moments: 

elec. dipole moment eR 



mag. dipole moment eh/Mc 



116 A Survey of Nuclear Reactions [Pt. VI 

It holds more generally. The absence of any J y = forbids zero-zero 
transitions completely, with one quantum emitted. For a transition 
in which a magnetic multipole fulfills the selection rules with the lowest 
value of multipole order L, the next higher electric multipole may con- 
tribute radiation less by an order of magnitude, since the reduction 
from order to order in L is estimated to be in the ratio (R/\) 2 < 
(p/Mc) 2 ~ (h/McR) 2 . If the lowest contributing multipole is of elec- 
tric type, the magnetic multipoles are negligible. 

1. Reduction of the Nuclear Dipole Moment. Classically, and even 
in atoms, the electric dipole transition is the most intense for radiation 
not too small in wavelength compared to the dimensions of the system. 
From our selection rules we could expect to see forbidden transitions, 
slow and non-dipole in character whenever large angular momenta have 
to be radiated. This is of course the origin of the well-known nuclear 
isomers, and of certain forbidden lines in the nebular atomic spectra. 
But there is a general argument which indicates that electric dipole 
transitions will not be so important, at least for nuclear gamma-rays 
up to some 10 Mev. 

The interaction, energy between a system of charges and the electro- 
magnetic field may be written : 

#int = 2] A-ji = X) A-(pVi) 

i i 

In the usual dipole approximation, we replace the operator v* by pi/m, 
and use the familiar relation between matrix operators, p = m di/dt. 
Then the dipole moment operator becomes 

Se t pi 

D = SejTi cc 

mi 

But plainly it is only the net displacement of the charges with respect 
to the center of mass which produces radiation from an isolated system. 
Then the operator whose matrix element between initial and final states 
determines the rate of radiation is 

P= X. — (X.--X) with AMX = MX> p + MX)x„ 

all particles TYli p n 

where M is the nucleon mass, for a nucleus with A nucleons, Z protons, 
and therefore N = A — Z neutrons. Here x p is the vector position of 
the pth proton, x„ that of the nth neutron, and X is the coordinate 



Sec. 9A] Radiative Processes in Nuclear Reactions 117 

vector of the center of mass, all in an arbitrary reference system. Now 
we can write 



1 

M 



! ( i -D? x "- e f? x " 



MA„ MA n 



(91) 

This is as though we calculated the dipole moment considering each 
nucleon with actual charge e* to have an effective charge only e,- — Ze/A. 
Clearly, if all the particles of the nucleus had e t -/m t - = e/M, there would 
be no electric dipole radiation whatever; the charge center and mass 
center would coincide, and the mass center clearly could not oscillate. 
In less extreme cases, where the motion of protons and neutrons is very 
similar — if they stick, say, more or less tightly together in alpha-particle 
sub-units — the electric dipole transitions would be at least much dimin- 
ished. There is no such general restriction upon the other multipoles. 
Indeed, magnetic multipoles will contain contributions both from the 
intrinsic magnetic moments of the nucleons and from other magnetiza- 
tions arising out of the transient currents coming from meson flow 
within the nucleus. These can be shown to be an inescapable conse- 
quence of exchange forces of various kinds (SI). In the deuteron 
photoeffect these currents are probably observable at low energy, where 
the major contribution is from a magnetic dipole transition. In general, 
then, we expect the magnetic multipoles to give somewhat larger con- 
tributions than the estimate made earlier would indicate, and we look 
for the electric dipole term to be much reduced, perhaps so much that 
the electric quadripole and magnetic dipole will overshadow its effects. 
2. The Sum Rules. This reduction of the dipole moment is limited 
by a very general result, an extension of the Thomas-Reiche-Kuhn sum 
rule long used in the atomic case. We can write for the integrated dipole 
absorption cross section 

r x 2^ 2 e 2 h , 

I <T a (Ey)dE y = — — — 2^ f on 
J Mc „ 

Now, independently of how the so-called oscillator strengths, /„„ = 
| X on fK n | 2 , vary with the energy difference between ground and ex- 
cited state, E n — E = h 2 /2MX „ 2 , the value of the sum is just 

M mmj/ei eA 2 

2-, Jon = 2-, — T~ \ J Wltn M any particle mass m; 

„ Sm,- i,j 2e \nii nij/ 

(92) 

provided only that the system absorbing the radiation consists of par- 
ticles of charge e,-, mass m,-, and that the forces between them are ordi- 



118 A Survey of Nuclear Reactions [Pt. VI 

nary forces. Exchange forces contribute a correction of similar size. 
Applying (92) to the nuclear case yields the result 

/•" 27rVh 1 / _ e 2 \ 7T 2 / h \ 2 , NZ 

I <r d (E y ) dE y = ( 2 23 — o I = ( ) m p<? 

J c A¥ p V t£2e 2 / 137 \M p cJ P A 

NZ 

= 0.058 Mev-barn (93) 

A 

From (92) it follows that a nucleus with N = Z, consisting of a set of 
infinitely well-bound alpha-particles, would have a strictly vanishing / 
sum. This is the result the previous section predicted. But, since 
actually we take the nucleus to contain not fundamental alpha-particle 
building blocks, but protons and neutrons, perhaps bound with finite 
forces into alpha-particle-like units, the integrated cross, section will be 
given by the form of (93). Taking the two results together, we can 
reconcile them by observing that the cross section will remain low, and 
we shall have little or no contribution to the / sum, as long as the energy 
of the gamma-ray is insufficient to break up any strong correlations into 
alpha-like structures. But the dipole transitions cannot really be 
prevented by any such internal binding of finite strength; they can 
merely be deferred. Sooner or later, as energy increases, transitions are 
made to states lying high enough so that any given sub-unit is broken 
up, and the / sum begins to grow, reaching finally the total given by 
(93), which any system of protons and neutrons must eventually show, 
whatever their internal motion. (We exclude exchange forces for the 
moment; they change nothing qualitatively.) 

From these general considerations we would expect that the gamma- 
ray transitions from nuclei would be electric dipole, magnetic dipole, or 
electric quadripole in most cases, with the last two having possibly 
somewhat higher probability, for transitions involving only a few Mev 
(Wl). But, for higher-energy gamma-absorption or -emission, say 
from 15 to 20 Mev — energies large enough to excite any transient con- 
figuration of nucleons, even the stable alpha-particle — the dipole transi- 
tions begin to show their deferred dominance and lead to integrated 
dipole absorption cross sections of the order of an Mev-barn (D5), for 
nuclei of middle A. This indeed seems to be a fair picture of what is 
still a murky experimental domain. 

Further extension of the sum rule type of calculation leads to rough 

information on the values of quantities like I a y {E y )E y n dE y ; these, 

taken with the experimental data (HI, K3), tend to confirm the picture 
of the last paragraphs at least roughly (L4). 



Sec. 9B] Radiative Processes in Nuclear Reactions 119 

B. Calculation of Radiation Widths 

The familiar formula for the rate of radiation of a quantum-mechanical 
system with dipole moment D radiating light of frequency v is 

Energy /second = -— | D»y | 
o c 

Introducing a self-evident notation for the type of multipole involved, 
and suppressing numerical factors near unity because of the roughness 
of our estimates, we may write the resulting width for electric dipole 
(2 x -pole) radiation: 

Bjit now we can estimate that, for a single particle moving in a region of 
radius R with charge e, the dipole moment is Z);/ = eR. For electric 
multipoles of successively higher order we can similarly estimate that, 
apart from numerical factors, the radiation intensity is reduced for each 
successive order in the ratio (R/X) 2 . Magnetic multipole moments are 
smaller than the electric moment of a given order by the factor (h/McR), 
which, using the nuclear radius value R = 1.5A^-10 -13 cm, leads to 

21 x 



[l-3-5---(2Z + l)] 2 

(94) 



1 /RV 

ec r^ f I 

— 137 \X/ 

p mag ^ J , ] ji elec r^ Q Q2^ — Mp.eleo 

— \McR/ ~ 

These estimates will be reasonable ones for the low-lying states of 
nuclei if the independent-particle, Hartree-like model is not too mis- 
leading. (We have suppressed factors depending on I, which would 
arise in a more consistent calculation from the complex angular behavior 
of the higher multipole radiation.) It seems not inappropriate to apply 
these formulas for the rough computation of those long-lived low-lying 
states of nuclei which are responsible for isomerism. The detailed 
comparison with experiment is beyond the scope of the present work; 
we make no correction for the additional non-radiative transitions due 
to internal conversion, and so on. But it is useful simply to show that 
the radiation widths we compute would give the possibility of lifetimes 
like those observed for reasonable values of radiated angular momentum 



120 A Survey of Nuclear Reactions [Pt. VI 

and energy. Table 9 shows lifetimes like those observed for the very 
transitions which the shell model predicts for the lowest states. 

TABLE 9 

Lifetimes for Radiative Decay of Low-Lying States 

A = 100 A = 200 

Multipole Energy = 200 kev Energy = 100 kev 

Electric 2 3 -pole . 029 sec 0.9 sec 

Magnetic 2 4 -pole 1 . 5 years 21 years 

We cannot expect the estimates of (94) to hold in the region where the 
level density is very high, and the idea of the excitation of a single 
particle no longer plausible. The sum rules themselves show that the 
estimate of a constant dipole moment, independent of energy, must fail ; 
the / sum would not even converge. A very rough idea of what to expect, 
consistent at least with the notion of a compound nucleus, may be gotten 
in this way. We expect the single-particle estimate to apply not to one 
single level of a highly excited nucleus, but to a whole group, dividing 
up the width, so to speak, among a great many levels each of which 
shares some part of the combining possibilities with the ground state. 
But, over how big an energy range must we spread the radiative width? 
A guess is afforded by the spacing of levels near the ground state, where 
it is not unreasonable to think that only a single particle has been 
excited. That spacing measures the energy region to be assigned to the 
excitation of one particle. Purely as a very rough orienting estimate, 
let us write for the radiative width of an average level in the region 
where levels are dense : 



1 /R\ 21 

r elec = _\ E x 

137 \X/ 7 



p(0) 



Ti™* = 0.02A~ 2A Ti 



elec 



P (E y )[l-3---(2l+l)] 2 

(95) 



where p(0) and p(E y ) are of course the number of levels per unit energy 
(really of levels with certain fixed J values) at excitation energies 
and E y . 

The actual radiation from any level formed in a nuclear reaction 
will of course involve transitions not simply to a fixed state below, but 
to all combining levels at lower energy. The total radiation width then 
for a given type of multipole radiation can be written 

r t ot' ,e ' ec = Z Tf^iEi - Ey) 

all/ 



Sec. 9C] Radiative Processes in Nuclear Reactions 



121 



Replacing the sum by an integral, and using the form of (95), we get 



l,e 






P(0) 



P\E y max) 



E _2l+l 



P(E. 



y max 



x 



Ey) ClEy 

(R/hc) 21 
137[l-3---(2Z+ l)] 2 



(96) 



The integrand represents the gamma-ray spectrum immediately follow- 
ing decay, without taking into account any of the subsequent cascade 
gamma-rays. The most common example of such radiation is that 
following thermal neutron capture. The spectra as measured (K7) 
show very marked effects of transitions to a few specific low-lying levels, 
such as the ground state itself, and do not fit our statistical estimates 
very well. There is not much information about the wide gamma-ray 
spectra which do seem to underlie these special lines and which presum- 
ably correspond to what we have estimated (D2). 

We tabulate, for nuclei in various regions of mass number, the com- 
puted radiation width following neutron capture, for radiation of various 
multipole types. The level densities are taken from (31), and the whole 
calculation is very rough. There seems here, too, to be evidence against 

TABLE 10 
Total Radiation Width after Neutron Capture: Various Multipoles 



A 


Max 

Ey = Bff 

(Mev) 


Widths in Electron Volts 


T\ elec 


p mag 


r 2 el 


J^mag 


1 observed 


60 
120 
200 


8 
7 
6 


4 

0.4 

0.1 


5 X 10-3 
0.3 X 10- 3 
0.07 X 10-3 


2 X lO- 3 
0.2 X lO" 3 
0.05 X 10" 6 


30 X 10 " 7 
2 X 10- 7 
0.3 X 10" 7 


0.2 



the full contribution of electric dipole radiation. Probably there are 
such transitions; some appear to have been identified in light nuclei, 
but they do not exceed in probability either magnetic dipole or electric 
quadripole by as much as an order of magnitude. 

C. Photo-Induced Reactions 

From the theory so far given, the behavior of gamma-ray-induced 
nuclear reactions follows in a somewhat sketchy way. The general 



122 A Survey of Nuclear Reactions [Pt. VI 

statistical theory describing the reaction as a two-step process is here 
applied : 

a y , p = S y (E)D p (97) 

where the cross section S y is the absorption cross section for the gamma- 
ray, and the factor D p , as in Eq. (40), describes the break-up, by emission 
of particle p, of the compound nucleus — here just an excited state of the 
target nucleus — exactly as in particle reactions. Below the binding en- 
ergy of a single neutron, no particle reaction can be observed, of course, 
since D p is exactly zero. Elastic and inelastic scattering of gamma-rays 
are possible, and would be governed by an analogue of the usual atomic 
dispersion theory, with the electric dipole moments perhaps not pre- 
dominant. As the particle threshold is crossed, D p rising from zero, the 
cross section <r yp rises as well, probably remaining rather small though 
increasing with a fairly high power of the energy, as a result of the im- 
portance of electric quadripole transitions. Only at energies where the 
excitation of the alpha sub-units becomes likely will the dipole moment 
assert itself; there the cross section S y rises to high values. As soon as a 
few Mev of excitation are available beyond the threshold for two- 
neutron, or even neutron-plus-proton, emission, this process will effec- 
tively eliminate the simple initial (7,^)- Thus a peak will be observed 
in the excitation function, with a width governed largely by competition. 
This peak, coming wholly from the factor D p , is superimposed on a 
fairly rapid rise and subsequent fall of the dipole cross section S y . There 
is still no clear division of the observed peak into the two factors; it is 
now fairly sure that the simple competition cannot be the entire reason 
for the width of a few Mev of the so-called "resonance" for the (y,n) 
reaction on a variety of targets (K4). The growth in neutron yield for 
a given gamma-energy, examined as a function of A, does, however, 
appear to be due mainly to the decreasing neutron binding energy and, 
hence, increased opportunity for two-neutron emission, as A grows. 

That the compound nuclear state formed by gamma-excitation might 
be of a rather special kind, with a particular internal motion, has been 
suggested (Cll, S10, T2). No clear evidence for this view has been 
produced which could not be duplicated on the present picture. The 

over-all { <r(E) dE, the presence of an apparent resonance due to com- 
petition, and the variation of neutron yield with A cannot distinguish 
the special motion from rather generalized features of any dipole ab- 
sorption. Good measurements of the peak widths and the relative 
yields of the various reactions are not yet at hand. 

That the assumption (97) is not entirely right seems demonstrated by 
the rather high yields of charged particles, especially of protons, from 



Sec. 10A] Nuclear Fission 123 

gamma-ray-excited nuclei. These yields are far in excess of those 
calculated from statistical theory whenever, as for medium or heavy 
nuclei, the statistical emission of protons is much reduced by the Cou- 
lomb barrier. The protons tend to be distributed in angle more trans- 
versely than isotropically. All of this suggests that a process which 
contributes only a small part of the total reaction — for medium nuclei, 
the protons are only a few percent at most — can occur in which protons 
are as frequent as or more frequent than neutrons. Any process which 
amounts to the leakage of the protons out of a small region of the nucleus 
before that region has lost its excitation energy by conduction of the 
"heat" to the whole nuclear volume will do. A direct photoeffect, in 
which the single proton is ejected from the smooth nuclear potential 
well, is an extreme case of this view (B15). Something between this 
and the statistical equilibrium idea is more likely to be correct. Emission 
of alphas, deuterons, and other fragments seems to give further signs 
of the need for a more detailed theory. 

Reactions may be induced by the varying electromagnetic field of a 
charged particle. This problem has been discussed in terms of the 
method of virtual quanta, mentioned in Section 8, mainly for electrons. 

SECTION 10. NUCLEAR FISSION 

No nuclear reaction type has been so much discussed, and none has 
attracted so many workers, 1 as the curious reaction called fission. The 
fact that this reaction involved so profound a rearrangement of nuclear 
matter that neutrons were emitted in greater numbers than one per 
divided nucleus permitted the chain reaction, and thus the large-scale 
release of nuclear energy, even under terrestrial conditions of pressure 
and temperature. We shall discuss fission rather sketchily and semi- 
quantitatively here, with major attention not to detailed experimental 
results, but to the features of most general interest. All information 
here presented, without exception, is from the published literature, 
much of it from the spate of studies of fission in the first two or three 
years after its discovery (T6). This section does not pretend to be a 
full guide to the present state of knowledge of fission. 

A. The Energetics of Fission 

The fission reaction cannot conveniently be written as we have written 
all others: T(z,p)R. The projectile may indeed be a proton or neutron, 
or the reaction may be observed to occur spontaneously, but the product 
nucleus is not one of the light nuclear particles, with a heavy residual 

1 See, for example, the semi-annual reports of the U. S. Atomic Energy Commission. 



124 



A Survey of Nuclear Reactions 



[Pt. VI 



nucleus. On the contrary, the products of fission are nuclei of sizable 
charge and mass, covering a range of scores of charge and mass numbers. 
In Fig. 18 we present the yield curve for several typical fission reactions 
(G3, Nl). The target nucleus has divided into two major fragments 



7 














U 235 + K 




^ 6 


— 






u 








0) 




J \Bi 209 +d 


1 lu 236 +w 








(3 

S 4 






JTh^l+a 


> 
"So 








* 3 

a 

e 

& 2 










O 


/ 








2 
£ 1 




7 


1 


ll 


\ 1 \ 



50 



100 
Mass number, .A 



150 



Fig. 18. Percentage of occurrence of product nuclear fragment of mass number A, 
plotted against mass number, in three different fission reactions. The neutron fission 
goes with slow neutrons; the alpha-induced, with alphas of 38 Mev; the deuteron 
fission, with 200-Mev deuterons. Note the trend toward a symmetrical distribution, 
and the invariably large spread in fragment masses. 

(with a few light particles sometimes also in a kind of spray). It is clear 
that the description of the emission of a particle p from an essentially 
unaffected residual nucleus R is inappropriate. 

But the phenomenon can be pictured in a very simple and convincing 
way from the most fundamental ideas of the nuclear model, treated in a 
nearly classical fashion. 1 We have throughout referred to the nuclear 
matter as a kind of "quantum liquid" of nearly constant density, some 

1 Bohr and Wheeler (B18) gave the first extended treatment of the theory; we 
follow them rather closely. 



Sec. 10A] Nuclear Fission 125 

1.45 X 10 14 g/cm 3 . A heavy nucleus is then a small nearly spherical 
drop of this fluid, with radius R = ro-A^ cm. In such a nucleus, look- 
ing apart from the entire complexity of quantum effects, odd-even 
regularities, etc., we can recognize a static equilibrium for the con- 
figuration. The attractive forces which cause the drop to cohere are, 
just as in a drop of water, short-range forces for any given nucleon satu- 
rated by the interaction with its nearest neighbors among the nucleons. 
In the roughest way, then, the binding energy of a nucleus of mass A, 
charge Z is just proportional to the number of nucleons present, each 
forming its full number of bonds with other nucleons. The negative of 
the binding energy, — E B , we define as usual as the total energy of the 
neutral atom (nucleus plus its surrounding electrons) minus the total 
energy of its constituent neutrons and protons when removed to large 
separations, including the energy of the electron needed to neutralize 
each proton. Then stability means positive Eb. Then Eb ~ volume, 
and, assuming constant density, the volume energy alone gives —Eb 
= —avA. But, like any liquid drop, the nucleus has a surface. Even 
for the heaviest nuclei, many nucleons lie on the surface of the drop. 
Their bonds are not all saturated; there is a net deficiency in binding 
energy, a positive surface energy exactly like the surface energy of a 
water droplet, but clearly of the greatest importance. We then expect 
— E B ~ —a (volume) + b (surface), under the same constant density 
assumption. There is yet another classically evident contribution to the 
nuclear energy. This is the mutual Coulomb repulsion of all the protons 
in the nucleus. It is not a saturating force, with finite bond numbers 
and a short range. On the contrary, it is a long-range force to which 
all the protons contribute. On the most naive assumption of con- 
stant charge density, the Coulomb energy of a spherical drop of radius 
R = r A ^ is simply the integral 

/Ze\ /Ze\ rrdVdV _3Z 2 e 2 3 ZV 

Vol/ ' W/ JJ I r — r' 1 ~5~R~~ 5r A* ^ 

sphere 

Purely classically, then, the binding energy of a liquid drop of constant 
density and uniform charge density 1 is given by the form 

Z 2 
-E B = -a v A + a s A % + a c — ^ (99) 

These simple ideas lead to a classical theory of fission. Suppose that 
now we have such a charged classical drop. If we slowly deform it by 

1 For a careful discussion of this whole procedure, and of the limitations of these 
ideas, see (F2). 



126 A Survey of Nuclear Reactions [Pt. VI 

elongating it, we clearly increase the surface area: this costs us surface 
energy. But the two halves of the drop are now found at a larger dis- 
tance from one another than in the spherical form. This means that 
their mutual Coulomb repulsion has decreased, and we have gained 
binding. As we deform the drop more, we increase surface energy and 
decrease Coulomb repulsion still more. Finally we can divide the drop 
into two remote spheres, each of half the original charge. In this state 
the Coulomb energy has been much reduced, and the surface increased. 
For a sufficiently high charge density, however, the Coulomb reduction 
must outweigh the surface energy increase, and the fissioned drops will 
be the stable configuration. The original sphere may then be in fact 
in a state of unstable equilibrium; any finite distortion from the sphere 
will cause the droplet to divide. Or it may be that, originally, small 
distortions into an ellipsoid will cost more surface energy than they gain 
in Coulomb energy, and the original sphere will be in a state of stable 
equilibrium. Yet finite displacements — distortion into something 
nearer a dumbbell shape — may so reduce the Coulomb energy that some 
intermediate distorted shape is a state of unstable equilibrium, capable 
of going either back to the original sphere, or on to two divided droplets. 
In this case the original drop is only relatively stable, and a genuine 
disturbance might cause fission. We shall see that this is the actual 
nuclear case at hand. 

1. The Semi-Empirical Theory. We can make all these notions more 
precise. Let us go back to the drop-model energy content appropriate 
for the constant-density nucleus. We wish to evaluate the constants 
for volume, surface, and Coulomb energy. In addition, we shall add 
two terms which represent an effort to make the drop idea conform more 
closely to nuclear experience. It is fundamental in nuclei of course that 
the exact numbers of individual nucleons have a real role, unlike the 
molecules in any large-scale drop, where small fluctuations in mass 
have an unimportant effect. There are two reasons for the effect, and 
both of them reflect the essentially quantum character of the nucleus: 
first, the total number of particles is small, so that quite generally small 
changes in A could have important effect; and, second, the strong inter- 
actions seem to result in something like the atomic shells showing as fine 
detail on the general course of the liquid-drop energy content. We 
shall throughout disregard the details of this phenomenon. There is 
a second consequence of the special nuclear forces : the protons and the 
neutron seem to tend to group in pairs or even in alpha-particle-like 
units. There is a binding energy bonus for having the neutrons and 
protons equal in number. We shall introduce semi-empirically, then, 
two terms into our binding energy formula: 



Sec. 10A] Nuclear Fission 127 

(1) A term, called the symmetry energy, which depends on the square 
of the difference between the number of protons Z and the number of 
neutrons N = A — Z. The fact that the term is quadratic represents 
the fit in first approximation to the trough of the valley in a binding 
energy surface, plotted against A and Z. The most stable isotopes lie 
in a crooked valley furrowing this surface; we imagine that the bottom 
of this valley can be approximated in cross section by a parabola. If 
we associate with each unpaired particle a definite energy, the whole 
term can be written 

(N - Zf a r 
(TV "t 6) A. 

(2) A term to stand for the last unpaired particle, expressing the fact 
that nuclei of even N and even Z are the most stable for a given A, and 
that those with N odd, Z odd are the least stable, while the odd-even 
or even-odd category is intermediate. This term may be empirically 
determined from the difference in energy between successive beta- 
transformations down a chain. With these additions, the semi-empirical 
formula becomes 

z * W - zf 

-E B = ~a r A + a s A V3 + a c ~ T .+ a r ± 8(A) (101a) 

A^ A 

We can evaluate ay and as by fitting the formula to the known mass- 
defect curve, which is by no means very accurate in the region above 
A = 40 or so. The procedure is made easier if we first evaluate a T in 
terms of a known a c . This we can do by fitting the valley of the stable 
isotopes, i.e., by finding the Z which corresponds to the most stable isobar 
for each A. The data fix this for odd Z at least to within about ±J^ 
for Z, making the valley pass between the two most abundant isobars. 
From our formula we need find only dM(A, Z)/dZ and set it equal to 
zero. This should mark the trough of the valley. [Note that M(A, Z) 
= -E B + AM n + Z(M H - M n ).] From dM/dZ = we get a relation 
between Z and A. The function so determined, called Z A , is given by 
the expression 

Z A _ (0.00081 + a T ) 

A ~ 2a T + 0.00125A % 

using ac = 0.584 Mev, as given below. If we smooth over the data, 
neglecting several kinks in the Z A function, we can find that, with ac 
as given, if a T = 0.083 mass unit, the fit to the empirical course of the 
curve is good. We consider throughout only nuclei above A ~ 20. A 



128 A Survey of Nuclear Reactions [Pt. VI 

set of good values for all the constants then is 

„ 77.3 I A \ 2 
-E B (Z,A) = -14.0A + 13.1A* + -7-\-z ~ z ) 



A 
A% lo;Aodd 



+ 0.584 — ^ + \ ±6V A ' Z even, A even (ioib) 



where we have expressed Eb in Mev. 

A similar semi-empirical formula due to Fermi has been extensively 
tabulated (M7) for nuclei of all plausible Z and A, going far beyond 
the known range. The form there used does not display the Coulomb 
energy directly, but combines it with the symmetry energy, using as a 
reference the bottom of the actual valley of stability Z A rather than the 
artificial situation of equal neutron and proton number. Written for 
the atomic mass in mass units, the Fermi form is 

M(A, Z) = 1.01464A + 0.014A^ - 0.041905Z A 

+ 0.041905(Z - Z A ) 2 /Z A 

^ |±0.036M«; ! ° dd ' V Ven nm , 
+ Z even, A even (101c) 

0; A odd 

with Z A /A = 1/(1.9807 + 0.01496A % ). An earlier version of the 
formula is cited in a more accessible place (Sll). Our (101b) is in 
reasonably good agreement with the Fermi formula, but the latter 
(101c) is preferable for actual calculations. (See Section 3B in Part IV 
of Volume I.) 

A more elaborate treatment has been given which includes a measure 
of the compressibility of nuclear matter, i.e., a departure from the con- 
stant density assumption (F2) . The effect is quite small. The Coulomb 
constant ac is not in fact evaluated from the semi-empirical formula for 
a large number of nuclei, but is given from the comparison of the so- 
called "mirror nuclei," isobars with N — Z = ±1, where direct meas- 
urement of beta-decay energy has been made (Fl, F3). The radius it 
corresponds to is somewhat larger than that we have taken from high- 
energy nuclear reactions; it gives r = 1.48 and not 1.4 X 10~ 13 cm. 
This difference may reflect in part the tendency for the protons to move 
to the outside of the nucleus, or it may simply reflect the inadequacy of 
the rather crude semi-empirical formula. 

Let us now consider the energetics of fission with formula (101b). A 
very simple model of fission may be made by assuming that the drop 
divides into two drops, each of the same charge density, proportional to 
Z/A. Let one fragment have charge fZ, the other (1 — f)Z, and simi- 



Sec. 10A] Nuclear Fission 129 

larly with the A' a. Then the energy released on such a fission, say 
AE(f), is given by the expression 

AE(f) = 13.1A^[1 -/« - (1 -/)«] 

Z 2 

+ 0.584 — [1 - f A - (1 - /)»*] in Mev (102) 

The volume and symmetry energies do not change; we neglect the small 
term 5 of (101), since we have not required integer values of Z or A. 
The treatment is entirely classical, and depends only on the assumed 
surface and Coulomb energies of drops of constant charge density. We 
plot in Fig. 19 the net energy release AE(f) as a function of / for nuclear 
drops of several charges. Note that, above a certain value of Z, fission 
will release energy for a wider and wider range of fragment sizes. For 
high enough Z, indeed, very small fragments — resembling alpha-par- 
ticles — may be released with a net gain in stability. This simple theory 
predicts the occurrence of fission and of alpha-radioactivity for high Z. 
Below the limiting Z value of course not fission, but combination of light 
nuclei, will represent a gain in stability. 

So far we have discussed only the energy difference between initial 
spherical drop and final spherical fragments. If this energy difference, 
AE(f), is positive, the process will proceed spontaneously. But the 
rate of fission, which will determine whether or not the process is ob- 
servable (rates corresponding to half-lives of more than Ty 2 ~ io 21-22 
years are not now observable), is still entirely open. In Fig. 20 we have 
represented quite schematically this complicated process. We have 
plotted the energy difference between initial spherical drop and dis- 
torted drop against a single parameter — some measure of the effective 
distance of charge separation. Of course the distortion can in fact be 
represented only by many parameters. In our simplified plot, the value 
AE is measured from the initial energy to the final energy, from E t to E f . 
We have plotted four cases: reference to the figure will make them clear. 
In case (1) fission is not an exothermic process and cannot occur. In 
cases 2, 3, and 4, it is exothermic and occurs spontaneously. In all of 
these, the initial sphere is in equilibrium at sp under the mutual action 
of surface and Coulomb forces. But in case 4 the equilibrium is un- 
stable; any finite disturbance will cause fission. The actual nucleus 
would last only a time comparable with the characteristic nuclear times, 
for the zero-point oscillations would cause the needed displacement. 
Cases 2 and 3 show initially stable nuclei; small displacements will not 
lead to fission, but will be followed by a return to the initial spherical 
equilibrium. Only finite displacement, adequate to supply energy greater 



130 



A Survey of Nuclear Reactions 



[Pt. VI 



=M(Z, A) 
-M(Z,fA) 




0.2 0.4 0.6 0.8 1.0 

Fractional charge of fragment, / 



1.2 



Fig. 19a. Potential energy classically available for fission into two fragments of the 
original charge density, but charges fZ and (l-f)Z, plotted as a function of fragment 
size, /, for several nuclei. For heavy nuclei, division into fragments of a wide range 
of sizes is exothermic; for nuclei lighter than, say, bromine, fission is endothermic 
even with the most favorable fragment size, and symmetric with / = 0.5. 



Sec. 10A] 



Nuclear Fission 



131 



than the height of the barrier, at P, will lead to fission. Classically, it 
will never occur spontaneously; the initial state is a; state of relative 
stable equilibrium. Quantum-mechanically, in the actual nuclear case, 
spontaneous fission can occur, by the tunneling effect — diffraction 
beneath the barrier. Whether or not this will occur at an observable 
rate will depend on both the height 
and width of the barrier. In case 2, 
the barrier is high and wide; we 
may expect that spontaneous fission 
will not occur observably, and that 
large energies of excitation compar- 
able with AE would need to be 
added to such a drop before fission 
could occur. This might be the case 
of a nucleus well beyond the limiting 
Z near 35, but not one of the natural 




radioactive series. 

It is evident that the most useful 
application of these ideas would be 
the calculation of the barrier heights, 
or fission threshold energies. The 
potential energy contour of the drop 
plotted as a function of some repre- 
sentative parameters determining its 
distortion from the sphere (of course 
two parameters are all we can plot) 
is a surface with a flat portion far 
away from the center (where the two 
fragments are far apart), rising to a 
fairly high elevation as the frag- 
ments approach and feel the Cou- 
lomb repulsion; in the very center 
of the contour map, where the drop is almost spherical, there is a 
crater-like bowl in the center of which lies the original sphere before 
fission. Small displacements will never cross the lip of the crater, but 
there are passes in the crater wall, the lowest of which lies above the 
crater center by just the fission threshold energy. This pass, or saddle 
point, is what we want to find, in respect to both its height and its 
position on the map, i.e., to the shape of the drop at the unstable equi- 
librium position, where it can either go on to fission or return to the 
original sphere. (See Fig. 21.) 



, 20 40 60 80 100 
Atomic number, Z 

Fig. 19b. Maximum potential energy 
classically available on fission into two 
drops of equal charge and mass, plotted 
as a function of atomic number Z. 



132 



A Survey of Nuclear Reactions 



[Pt. VI 




50 100 

Distance apart of nuclei centers (10" 13 cm) 

Fig. 20. Potential energy of nuclear drop as a function of distortion. The compli- 
cated distortion is here represented by a single parameter, some measure of effective 
distance of charge separation. The energy surface is plotted in the plane which 
represents the most favorable distortion, through the "pass" in the energy crater. 
Note the stable position at sp, the original spherical shape, the increase in potential 
energy to reach the pass at P, the energy threshold for fission, Eth, and the long 
decline down the curve to D and beyond. The distant part of the curves is simply 
the 1/R behavior of the Coulomb energy of spherical charges. If continued in until 
the fragments touched, any distortion being neglected, this would reach the points 
marked C, the end of the Coulomb barrier. How the drop distortion allows fission 
at much lower thresholds is seen from the bending over before point C. 



Sec. 10A] 



Nuclear Fission 



133 




134 



A Survey of Nuclear Reactions 



[Pt. VI 



The full calculation is of course difficult. The number of degrees of 
freedom of a classical charged drop under axially symmetric distortions 
is very great, but some important consequences can be obtained quickly. 

(1) Let us consider the limiting case, where the charge density is 
small and the surface energy far outweighs the Coulomb repulsion. In 
this case, the drop will try to minimize its surface at all deformations, 
and the spherical shape will be stable for every distortion up to the very 
radical one of allowing the drop to be molded into two spheres and mov- 




Undistorted 



a; = 0.74 




(a) 



(b) 



x = 0.65 



x«l 





(c) 



(d) 



Fig. 22. A series of equilibrium shapes for the nuclear drop. Shape (a) is of course 
the original sphere; (b) the shape of the drop at the critical distortion for 
x = 0.74 = (Z 2 /A)/(Z 2 /A)n m ; (c) critical shape for x = 0.65; (d) the shape corre- 
sponding to a: « 1, division into two nearly tangent spheres, where the Coulomb 
tidal distortion is neglected (low Z). 



ing those spheres just out of the range of the surface forces. The shape 
of Fig. 22d is the case we shall compute. Here the Coulomb energy is not 
strictly zero, but only very small. The critical shape will then still 
allow for a tiny neck connecting the two fragment drops, which can be 
torn apart by the Coulomb repulsion. If we neglect the energy of the 
neck, which is very small in our case, we can simply compute the differ- 
ence between the energies of the two equal spherical fragments and the 
original sphere. The fission threshold energy, E t h, will in this case be 



E th = 2-4.Tr 



[©"*] 



M(A) 1A r ] 2 <r + 2 



3 (Ze/2) 2 



+ 



5(A/2)^ 
(Ze/2) 2 3 (Ze) 2 



2(A/2y A r 5A A r 



(103) 



Sec. 10A] Nuclear Fission 135 

(with <r = nuclear surface energy/cm 2 ) from which 

E th Z 2 e 2 /r 

= 0.260 - 0.065 -f- (104) 



MA H r ) 2 <r ' ' A iirr 2 a 

e 2 /r Z 2 _ 5 Coulomb energy 

where the parameter — • — is just - • . 

4irr a A 3 Surface energy 

(2) Now we consider the other limiting case, with the original droplet 
so packed with charge that the slightest displacement from the spherical 
shape will produce fission. Here Eth = 0. To compute this we con- 
sider small constant-volume distortions, axially symmetric, without mo- 
tion of the center of mass, but otherwise arbitrary. We may represent 
the radius of the drop surface as a function of the co-latitude angle, 0, by 
a series in the orthonormal set of Legendre polynomials 

y(6) 

= 1 + 0-P x (cos 9) + a 2 P 2 (cos 0) + a 3 Ps(cos 0) -\ (105) 

R 

Here the coefficient of Pi (cos 8), oti, is set equal to zero, since for small 
displacements (where the higher harmonics may be expected to grow 
small), it corresponds to displacement of the center of mass. One con- 
dition is set on the motion by this requirement. The purely mathe- 
matical requirement that the function y(6) be single-valued for repre- 
sentation as a Legendre polynomial expansion does restrict the physical 
motion somewhat. Such a shape as that of the other limiting case — 
two near-spheres joined by a thin thread — cannot be so represented. 
This turns out to be an unimportant restriction; our answer is very far 
from such a shape for the observed fissioning nuclei. 

If we make the calculation of the classical area and Coulomb energy 
with assumption (105), and keep only the leading term, in a 2 , we obtain 
these results for surface and Coulomb energies of an infinitesimally dis- 
torted drop (81), neglecting all terms of higher order in the small co- 
efficients a n : 



5A H r \5 / 



tfsurf - -E surf sphere = 4x4 Hr„ a , 



to . - 

(106) 



isurf - ^surf •" " = 4irA "ro- 
Then the fission threshold energy, measured in units of the undistorted 
surface energy, becomes 

+ ■■■) (107) 



±irA % l0 2 c 5 V # surf sphere 



136 



A Survey of Nuclear Reactions 



[Pt. VI 



We can read off the result: any displacement will lead to fission (i.e., 
Eth < 0) if JV phere /#surf Sphere > 2. Writing this result in terms of the 
semi-empirical constants, we have 

E c /E sm! | lim = Z 2 /A \ iim -a c /a s = 2 .-. Z 2 /A | lim = 45.0 (108) 

0.30r- 



0.20 — 






0.10 




o.e l.o 

x =E c /2E a =CZ 2 /A)/(zVA) lim 



Fig. 23. The fission threshold energy, Eth, as a function of the parameter x. The 
dimensionless function /(x) is just Eth measured in units of the nuclear surface energy 
in its undistorted spherical form, and x is defined by the relations x = Ec°/2E S ° 
= Z 2 /A/(Z 2 /A)um- From fitting the experimental data for U 239 , we get 



(Z 2 /A) v , 



■ 48 (see text). The function f*(x) is the energy difference between the 



sphere and the shape of Fig. 22d, two equal nearly tangent fragments, again meas- 
ured in units of the surface energy. For small x, below the vertical stroke on the 
curve of f(x), the function f(x) is taken from its limiting value for small Z; above 
the right-hand vertical mark, f(x) has been calculated numerically. Between the 
two strokes f(x) has been simply interpolated free-hand. 

We can of course continue the series still further, and examine how 
Eth/E aur f spheie , which we will write as f(x), behaves as a function of 
x = Ec/2E S . This has been done out to quite high powers of the small 



Sec. 10A] Nuclear Fission 137 

coefficients — with as many as six or eight harmonic terms. A numerical 
calculation, done with high-speed calculators (Fll), has been carried 
out with ten harmonic terms, and without making any power-series 
expansion in the coefficients. The function fix) is known near x = 
from the calculation in case 1 above, and near the limit x = 1 we have 
plotted the calculated results in Fig. 23. The dashed part of the curve 
is a reasonable interpolation. We can regard x either from a theoretical 
point of view, as the ratio Ec/2E S , or in a form more directly comparable 
with experiment, as (Z 2 /A)/(Z 2 /A) ]inl . In the latter form, we see that 
we can read off the value of the fission threshold E t h for any nucleus by 
computing the surface energy asA 2/3 , noting that E t h = f(x) ■ as A y *. 

We can now apply these results best not by taking the limiting 
{Z 2 /A\ Ta entirely from our semi-empirical constants, but by fitting 
one observed fission threshold (S8) to the fix) curve. If we fit E t h (U 239 ) 
= 6.67, and take, for the surface energy for U 239 , 502 Mev, we obtain 
f[x(U 239 )] = 0.0132. This yields Z 2 /A\ Um = 47.8, which can be com- 
pared to 45.0, obtained by taking the constants directly [Eq. (108)]. 

2. Quantum Effects. We have treated the problem wholly classically. 
It should be observed that this neglects two quantum-mechanical effects. 
First, fission may take place for excitation energies below the fission 
threshold by reason of the tunneling effect; if the representative point 
comes near the pass over the crater lip, it will often go through the thin 
barrier remaining. In addition, the vibration of the drop in the dis- 
torted mode will have a zero-point energy. Both of these effects can be 
estimated by computing how the barrier looks as the representative 
point moves in its path over the energy surface. This can be done under 
the assumption that the drop distorts through the sequence of shapes 
near the saddle-points for larger values of x. With this assumption, 
Frenkel and Metropolis (Fll) calculated the barrier as a function of 
distortion. The Gamow penetration could then be estimated by using 
this potential barrier, and an effective mass estimated from hydro- 
dynamic arguments. They obtained a penetration probability P for 
various E th given by P = io -7 - 85 ^. The zero-point energy can be 
calculated from the shape of the crater near zero distortion. This gives 
a result of some 0.4 Mev for the uranium case and leads to an estimate 
of the oscillation frequency. Such a zero-point energy, small compared 
to the excitation involved, implies that the motion can be represented 
classically as a rather well-defined trajectory by building up wave 
packets from the actual quantum states. The threshold estimate is 
then made as follows : 

-"th (Classical) = -£/ z ero-point l -* neutron ~T~ -'-'neutron binding 



138 A Survey of Nuclear Reactions [Pt. VI 

Using the results that neutron binding in U 239 is 5.2 Mev (from the 
semi-empirical formulas) and that the measured neutron kinetic energy 
is 1.0 Mev, we get the 6.6 Mev used above for E t h (classical). 

Only qualitative results can be expected from this theory. The 
actual fission is not into symmetric fragments at these excitations, 
and the fission thresholds do not in detail vary from nucleus to nucleus 
in the smooth way here predicted. Moreover, the spontaneous fission 
rates are in poor agreement. It must be observed that the very large 
exponents which occur mean that a small error in the barrier will lead 
to enormous errors in the rate calculations. Perhaps only the logarithm 
of the decay rate should be expected to be of some significance, within 
a small factor of the truth. Arguing the other way is more sure. It 
turns out from the f(x) curve (it will be an interesting exercise for the 
reader) that for Z > 98 the typical isotope would be radioactive to such 
an extent that it would probably disappear from the earth in geologic 
time, not because of the accidents of alpha-particle disintegration rates 
(which vary both up and down in just a few Z or A units), but because 
of the inescapable effects of fission. Thus the fact that the heaviest 
nucleus occurring in nature is uranium is a consequence only of the 
kinks in the binding energy curve, coming from the closed shells at 
Z — 82, N = 126; this means that the next few elements have short lives 
against alpha-decay. But the more fundamental reason that no ele- 
ments of Z = 100 or up are found is certainly fission. It will be inter- 
esting to see the first nuclear species whose lifetime is controlled by 
spontaneous fission. 

There is one more deduction from this simple theory which is of inter- 
est. The calculation of Eq. (104), which we can plot as the function 
f*(x) for < x < 1 (see Fig. 23), gives the energy difference between 
the original sphere and the nuclear matter arranged as two equal frag- 
ment spheres just touching. Now the energy released on symmetrical 
fission is a given function of Z and A, independent, of course, of the 
path followed in dividing. If the released energy is Ai?(3^), then the 
energy difference between the critical distorted shape and the separated 
fragments is just f(x)asA ^, while that between separated fragments and 
two tangent fragment spheres is f' t (x)asA 2A , which is, from Fig. 23, some 
25 or 35 Mev greater in the region of the fissionable nuclei. This means 
that, when the distorted surface finally tears to release the two frag- 
ments, which when far away will eventually come to their stable spherical 
shape, the difference of about 30 Mev is stored as energy of distortion — 
energy of excitation — in the two fragments. It is, moreover, plain that 
the reverse of fission — union — will in general require much more kinetic 



Sec. 10B] Nuclear Fission 139 

energy than that released in fission. It will be very unlikely that the 
special distortion which leads over the low pass in the crater lip will be 
achieved in the random collisions of nuclei. 

B. The Products of Fission 

The most obvious consequences of the simple theory given above are 
in fact not observed. The fission fragments do not appear as equal 
nuclei with / = }/%. On the contrary, the distributions of Fig. 18 are 
those observed experimentally. No clear explanation of this fact has 
yet been given. It seems almost sure that there is no complex distortion 
type which leads to a low-lying pass for some asymmetric distortion of 
the classical drop. It is much more probable that again we have to do 
with a reflection of the nuclear shell structure, permitting some motions, 
some ratios of charge to mass in division, more easily than others. The 
closed-shell N's at 50 and 82 do seem to mark near-maximum yields. 
There is some evidence that isotopes of those stable elements with 
highest N — Z are favored, but the expected predominance of even 
products is not yet found in the admittedly complex chains of fission 
fragments. It should be obvious by now that the initial products of 
fission are not in general stable nuclei, but are very neutron-rich, and 
begin chains of beta-decays to reach the appropriate ratio of Z/A for 
their smaller A (B18, W4). It has been shown that, in this beta-decay 
chain, excitations are in some cases in excess of the neutron binding 
energies for the product nucleus of a given beta-decay. This leads to 
the emission of delayed neutrons whose time of emission after fission is 
determined by the preceding beta-decay processes. Much more numer- 
ous are the so-called prompt neutrons, which emerge after a time short 
compared to any possible beta-decay. It seems very likely that these 
neutrons are evaporated from the highly excited fragments as they fly 
apart after the fission act. This would imply that some two or three 
neutrons are emitted, and that their energy spectrum and angular dis- 
tribution would be the quasi-Maxwellian one of our statistical theory 
(compare Fig. 6). The fission fragments, however, continue in motion 
at their high speeds of near 1 Mev/nucleon for a distance of a few milli- 
grams per square centimeter, or some 10~ 3 cm, in solid matter. The 
evaporation will occur with neutron widths measured in many kev at 
least, or in times short compared to the fragment time of flight. So 
the neutrons will evaporate isotropically from the moving fragment, 
and in the laboratory system will be emitted preferentially forward, 
since their kinetic energy will be quite comparable to that of the frag- 
ment, expressed as Mev per nucleon. Some evidence has been obtained 
to confirm this picture as well (W9). 



140 A Survey of Nuclear Reactions [Pt. VI 

The products of fission, then, are mainly the two large fragments and 
the prompt neutrons. The fragment decay leads to beta-particle chains, 
gamma-rays, and the delayed neutrons. But there are other charged 
particles which emerge at least occasionally upon fission. In about one 
fission in five hundred an alpha-particle forms the third fragment, with 
kinetic energy from a few to about 25 Mev (M2). It seems well estab- 
lished that these smaller fragments actually originate from the com- 
pound nucleus, and from a tendency to come off at right angles to the 
direction of the main fragments, and that they arise during the act of 
fission. 

The systematic change in shape of the fission fragment size-distribu- 
tion curve with energy of the bombarding particle is the most striking 
feature of Fig. 18. Apparently, as the excitation exceeds the fission 
threshold more and more, the tendency toward the symmetrical fission 
expected in the simple theory becomes stronger. Slow-neutron uranium 
fission is never symmetrical ; at energies around 100 Mev, the symmetri- 
cal division is the most probable. Direct measurement of the ionization 
energies of the fragments confirms the observation on the masses of the 
fragments (J4). Whatever special effects produce the asymmetry are 
increasingly unimportant as more energy becomes available to the 
vibrating drop. It does not seem true, however, that the fragments of 
an asymmetric division have total fission product energies higher than 
the fragments of nearly symmetric division; there is even some tendency 
toward the reverse. 

C. Fission Cross Sections 

Fission, as a mode of decay of a compound nucleus, should be produced 
by any means that excited the nucleus above the required threshold 
energy. This has been verified for slow and fast neutrons, for protons, 
deuterons, and for alpha- and gamma-rays. In general the cross section 
can be computed by our usual methods — simply writing the cross section 
for formation of the excited compound nucleus by a particle of energy 
E v , and following it by the factor for its competitive disintegration: 

i{W)YAW) 
*,(E P ) = ,(E P ) " ' " > (109) 

*/ T An "T J- 7 T ' ' ' 

where 1/ is the width for fission from a compound nucleus of excitation 
W. The calculation of the fission width is the calculation of how the 
excitation of the nuclear drop is distributed among all its possible modes 
of motion: there is available a large amount of phase space for motions 
which lead to distortions from which the drop cannot divide; only a 
small volume in phase is assigned to such motions of the point repre- 



Sec. 11] Nuclear Reactions at High Energy 141 

senting all the parameters of distortion as do lead over the pass in the 
wall of the energy crater. Below the threshold for fission, the excitation 
energy can be lost by any other energetically available means, usually 
gamma-radiation or neutron evaporation, but not by fission. As the 
excitation W = E p + Eb(p) increases to near the classical fission thresh- 
old, tunnel effect can lead to fission. The fission width will increase, 
and fission will become a competitive reaction. We can expect the 
fission width to rise rapidly, perhaps something like exponentially, as 
the fission threshold is considerably exceeded. What the cross sections 
will be then will depend largely on how high the fission threshold is 
relative to the excitation energy of a given mode of excitation. Thus 
<rf(E n ) for U 238 + n 1 is zero until a threshold of about 1 Mev, from 
which it rises slowly to a value comparable with irR 2 . It stays nearly 
constant because for some time 1/ and r„ rise about in the same way. 
We know from the experiments with very fast fission, on elements like 
bismuth, where the fission threshold is some 12 Mev or more, that the 
neutron specific width rises more rapidly than that for fission. Very 
high excitation in Bi 209 leads mainly to multiple neutron evaporation; 
some ten or twelve neutrons are boiled off before fission becomes a very 
important competitor. Only when the fission threshold has come down, 
because of the gradually increasing value of Z 2 /A in this neutron boiling- 
off, does a-/ begin to grow to a value comparable with unity, actually 
about 0.1 or 0.2. This is confirmed both by the fission-product dis- 
tribution (which is that of fission by a nucleus of A ~ 200) and by the 
fact that the fission fragments come apart with energy near that ex- 
pected from fission near the threshold energy. The distribution of 
fragment sizes becomes broader with increasing excitation; the mode of 
division does not need to be so special. Even so, it is plausible that there 
are many more ways for a highly excited drop to emit neutrons than to 
divide by fission, so that, while the I> rises rapidly, it does not rise so 
rapidly as the other terms of total r, almost all of which, at these ener- 
gies, arise from neutron evaporation. 

Special processes, like photofission and fission in the region of applica- 
tion of the dispersion formulas, have been discussed in the literature 
(B18, G3, Nl). 

SECTION 11. NUCLEAR REACTIONS AT HIGH ENERGY 

The processes we have discussed so far are characterized by the fact 
that the energy of the incoming nucleon, both potential and kinetic, 
becomes quickly shared among all the particles of the nucleus. After 
a relatively very long time, during which numerous nucleon collisions 



142 A Survey of Nuclear Reactions [Pt. VI 

occur, this energy may again become concentrated in a particular 
nucleon, or in some special mode of motion, and the nucleus will de- 
excite by particle emission, fission, or the slow processes of radiation. 
As the energy brought in grows, the number of available modes of 
disintegration naturally increases, and the lifetime of the compound 
nucleus becomes shorter and shorter. With energies in the range up to 
some 30 to 50 Mev, the nucleus will emit typically several nucleons 
before it cools, and the reactions are increasingly complex. Such reac- 
tions as (p,3n) and (p,4n) have been studied (K5). But, as the energy 
increases, there is a gradual transition to a state of affairs which is best 
described in a conceptual framework quite the opposite from that 
suited to the compound nucleus. This high-energy point of view works 
best above some hundreds of Mev of incoming nucleon energy, and the 
two schemes will merge in the broad transition region between. 

Most characteristic of the high-energy region is the fact that the mean 
free path, in nuclear matter, of the fast nucleon is comparable with the 
radius of the nucleus itself. The nuclear matter is no longer a black, 
impenetrable obstacle, which traps the incoming particle, but a kind of 
sphere of dilute gas, through which the fast-moving nucleon has an 
appreciable chance to penetrate without any collision whatever. Even 
a collision or two do not by any means rob the incoming nucleon of most 
of its energy. The momentum transfer in such a collision, which is 
typically transverse to the classically pretty well-defined trajectory of 
the short-wavelength incident nucleon, is of the order Ap ~ h/R, where 
R is the range of nuclear forces (S7). This implies an energy transfer 
of some 30 Mev. In the short time of such a collision (~10 -22 sec) 
the struck nucleon will make only very few, if any, collisions with the 
neighboring nucleons of the nucleus, and, although the momentum trans- 
fer is not so large that we are wholly safe in regarding the struck nucleons 
as free, still that will be a reasonable initial approximation. We shall 
see that the principal effects of the nucleon environment are conse- 
quences of the Fermi degeneracy of the nucleon matter. The collision 
will be modified by the impossibility of leaving the struck nucleon in a 
momentum state already occupied by some other nucleon of the nucleus. 

The geometry of the approach of the nucleon will determine the se- 
quence of events. If it passes through the nuclear edge, it may make 
no collision, or perhaps one. A very fast nucleon, having lost only 30 
Mev, will emerge. The nucleus left behind may be excited by the full 
amount of the energy given to the struck nucleon, or by very much less 
if the initial path of the struck nucleon allowed it as well to leave the 
nearby nuclear surface without further collisions. A fast particle striking 
Jhe very center of the nucleus, where the nucleus is thickest, may have 



Sec. 11 A] Nuclear Reactions at High Energy 143 

to travel a few mean paths in nuclear matter, and will typically make 
several collisions, leaving in its wake a few nucleons of 20 to 30 Mev 
energy each, perhaps losing all its energy, being finally captured. The 
secondary nucleons in turn can escape only if they are close to the 
nuclear surface and moving outwards. Some may do so; others will 
collide with many nuclear particles, and gradually spread their energy 
around among many modes of motion, forming a heated compound 
nucleus, which will cool off just as in the reactions of lower energy. A 
typical many-particle nuclear reaction, often called, from its appearance 
in nuclear emulsion photographs or in the cloud chamber, a "star," can 
then be thought of as a kind of approach to thermal equilibrium, be- 
ginning with a well-defined single-particle trajectory, followed through 
a cascade of secondary, tertiary, etc., collisions, and ultimately resulting 
in a diffusion of excitation energy like heat through a conducting sphere. 
The initial stages form the high-energy limit; here the reaction is to 
be described by a step-by-step following out of the nuclear cascade, 
taking into account collisions with one or with a few correlated nucleons; 
finally, the energy is shared by the collective motion, in something like 
transient thermal equilibrium, which is the low-energy, compound 
nucleus picture. A single star event may exhibit both features; it is the 
fact that the first stage takes a time ~10 -22 sec, very much less than 
the 10~ 16 sec of the last stage, which permits a more or less sharp division 
into the two stages. On this sharp division, which is only approximate, 
the simplicity of the picture depends. The intermediate time, during 
which the energy is distributed among too many degrees of freedom 
to be described in detail, but too few to approach any sort of quasi- 
equilibrium, is beyond the present theory; such complex cases will be 
especially important in the lighter nuclei, and for a mass A nucleus at 
energies 10A Mev, enough to dissociate the whole nucleus. Our treat- 
ment will deal wholly with the two well-separated stages: the initial 
cascade and the final thermal equilibrium. 

A. The Nuclear Cascade 

The initial cascade has been followed out by the most direct means. 
The theory has been the semi-empirical one of the so-called Monte 
Carlo method (G5), in which after the establishment of the model and 
the appropriate cross sections— which are the decisive features for the 
accuracy of the results— the calculation is done by considering the 
successive events in the motion of the incoming nucleon and all its 
collision partners, with their collisions in turn; choosing the actual steps 
by a series of random choices; and finally cutting off the whole process 
when some arbitrary low-energy limit is reached beyond which the energy 



144 



A Survey of Nuclear Reactions 



[Pt. VI 



is assigned as equilibrium excitation of the compound nucleus. This 
step-by-step process is extremely tedious, and only a few hundred indi- 
vidual "stars" have been followed through on paper. The results are 
subject then to a fair degree of statistical uncertainty over and beyond 
shortcomings of the model. But enough has been done to make the 
general features of the high-energy stage in star formation fairly clear. 
1. Fermi Gas Model of the Initial Nucleus. The short collision time 
and high recoil energy make most plausible the use of the simplest of 















V 


J 




Fig. 24. Diagram of the potential well in which the nucleons are bound in the sim- 
ple Fermi-gas model. The Coulomb barrier affects only the protons. The well 
depth is determined by the nuclear density and the assigned binding energy. 

nuclear models, that of the Fermi gas of non-interacting nucleons con- 
tained in a well of assigned diameter and depth. Naturally the correla- 
tions actually arising from the nuclear forces, here ignored, will be 
important eventually. One clear sign of this is found in the process of 
the pick-up of a nucleon from the nucleus, to emerge bound to another 
outgoing nucleon in an emitted deuteron, a process far more important 
than the Fermi gas model would predict. But, aside from such details, 
it is appropriate to begin with a Fermi gas. 

The procedure is the familiar statistical one for finding the distribution 
of a completely degenerate Fermi gas, at T = 0°. The nucleus is re- 
placed by a spherical potential well in which the particles move without 
interacting. Each particle has a wave function which is simply a plane 
wave; the finite depth of the well, and hence the leaking of the plane 
waves into the classically forbidden region outside the well, is neglected. 
Then a particle of each spin is placed in every momentum state until 
all particles are accounted for. The neutrons and the protons are re- 
garded as entirely independent and as moving in separate wells; the 
additional Coulomb barrier for the protons may be regarded as beginning 
at the well edge. Figure 24 shows the model schematically. Replacing 



Sec. 11 A] 



Nuclear Reactions at High Energy 



145 



the sum over the filled states by an integral, and remembering that 
(2s + 1) particles with different spin orientations may be placed in each 
cell in phase space, of volume (2xh) 3 , we obtain for the number of 
particles of a given kind, with spin s, in a sphere of volume V, 



N s 






(2s + l)p 2 dp cK2/(2xh) 3 



where p F is the limiting momentum to which the states are occupied, 
and the energy corresponding to pf, the Fermi energy E F , is given by 
E F = p F 2 /2M a . If we take the sphere of nuclear matter to be a sphere 
of constant density independent of the mass number A, as is usual, and 
write for the radius of the nucleus R = r -A H , we can easily find an 
expression for the Fermi energy for the nucleons of one kind: E Ft = 
(18tt) k /8 X h 2 /Mr 2 ■ (N s /A) H , where N s is either the atomic number Z 
for the protons, or the neutron number N = A — Z for the neutrons. 
This gives the numerical result: E Ft = 9.7(N s /A) 2A (a /r ) 2 Mev, where 
we have written a /r , the ratio of a , the classical electron radius, 
2.82 X 10~ 13 cm, to the nuclear radius parameter. For orientation a 
few values are listed in Table 11. The well depth is now fixed to give the 

TABLE 11 
Proton and Neutron Fermi Energies for Various Nuclei 



Ef (protons) 
Ei?(neutrons) 



Cu 65 



n> 



1.4 
22.6 
26 



1.6 

17.5 
20 



Cs 1: 



1.4 
22 

27 



1.6 
16.9 
20.7 



1.4 
21 

28 



1.6 X 10" 1S 
16.1 Mev 
21.5 



highest-lying neutron state about 8 Mev binding energy, and an addi- 
tional Coulomb barrier may be assigned for the protons. This implies a 
nuclear potential well for neutrons in lead about 35 Mev deep, using 
the smaller value of the radius parameter, and an additional Coulomb 
barrier V c for the protons about 0.5(Z/A M )(a Ao) Mev high, or <~14 
Mev for lead. In the calculations the two types of nucleons are often 
represented by a single gas of about average properties. 

In the region of high energies, the simple Fermi gas model has two 
characteristic consequences which are certainly to be expected even from 
more realistic models of the internal nucleon motion. It is perfectly 
clear that the complete neglect of nucleon interactions is an extreme 
procedure which tends to underestimate the range of momenta repre- 



146 A Survey of Nuclear Reactions [Pt. VI 

sented in the statistical motion of the nucleons. One can regard this 
lack of a high-momentum tail as implying either the neglect of spatial 
correlations, in which the nuclear density fluctuates toward higher 
values because of the nucleon interaction and the transient formation 
of sub-units of two, three, or four nucleons, or as a neglect of occasional 
transitions, made possible by nucleon interaction, from the lowest 
states of the Fermi gas to some of the higher unoccupied momentum 
states. Both ways amount to the same thing; they form the basis for 
regarding the Fermi gas calculation as a first step in a systematic approxi- 
mation method. We shall look here, however, into the consequences 
of the raw Fermi model as they affect high-energy interaction events. 

(a) Influence of the Exclusion Principle. In this model, the incoming 
nucleon, once it enters the nuclear volume, fills a hitherto unoccupied 
momentum state. Now it enters into a collision with some one of the 
nucleons already present, lying in one of the filled levels below the 
limiting Fermi momentum p F . The spirit of the model, representing 
all nucleons as free, now clearly requires that the momenta of the two 
collision partners after the collision lie in an unoccupied region of the 
momentum space. A simple exchange of momenta corresponds to an 
elastic collision. Any other event must begin by fulfilling the condition 
of entering fresh momentum space regions. Evidently a small momen- 
tum transfer to the struck nucleon is forbidden unless the nucleon ini- 
tially lies very near the edge of the sphere in momentum space of radius 
p F . As long as momentum transfers not very great compared to p F are 
of importance in the process, which means that the incident particle 
may have momentum up to many hundreds of Mev/c (or even con- 
siderably higher if forward scattering alone is under consideration), 
the cross section for collisions with the average nucleon of the Fermi gas 
will be much reduced by the requirements of the exclusion principle. 
This will evidently increase the chance of passing through nuclear matter 
without collisions, and will tend to reduce the relative probability of 
small-angle scattering, both near zero degrees in the center-of-mass 
frame and, because the projectile also satisfies the exclusion principle, 
near 180° as well. 

Figure 25 presents the geometrical considerations in momentum space. 
A calculation (G5) on this purely geometrical basis shows that the 
exclusion principle reduces an assumed isotropic cross section for scatter- 
ing in the ratio: , - 2 \ 

_ / 7 PF 2 \ 

0"av — Cf r ee II—- — - 1 

\ 5 pi 2 / 
where o- free is the isotropic scattering cross section per unit solid angle for 
a free nucleon, and p t is the incident nucleon momentum. This formula 



Sec. 11 A] Nuclear Reactions at High Energy 147 

holds only for values of p t > 2 V2 p F ; below that incident momentum the 
result is more complicated. The direct analytical evaluation has not 
yet been performed for a case with the cross section anisotropic. The 
angular distribution can be approached through the same geometrical 
scheme, if the scattering cross section is taken as a function of the mo- 
mentum transfer | p { - p f | = | 5 | only, which corresponds to the Born 
approximation without exchange. 




Fig. 25. Kinematics of the collision between an incoming nucleon and the nucleons 
of the Fermi gas. The incident, recoil, and final momentum vectors are shown, with 
spheres of radius in momentum space just p F , the Fermi momentum. The shaded 
region is excluded; no collision in which the vectors Precou or p flna i terminate within 
this region is permitted by the Pauli principle. 

(b) Influence of the Zero-Point Motion. The angular distribution of 
particles scattering from the Fermi gas involves two distinct and im- 
portant effects: the exclusion of low-momentum transfer collisions, and 
the smearing out of the otherwise unique correlation between scattering 
angle and energy of emerging particle, which can be regarded as the 
effect of the zero-point motion of the struck nucleons of the Fermi gas, 
which of course spread over a wide range of momenta, and hence of 
relative velocities, as seen from an incoming nucleon of fixed energy. 
The geometry of the event is just that of Fig. 25, and one can write 
rather complicated expressions for the cross section in the laboratory 
system for scattering a particle into the element of final momentum of the 
scattered particle, p/ - . 

¥- = 'WtPF, P/, 6) (HO) 



148 



A Survey of Nuclear Reactions 



[Pt. VI 



where/ is a function of the momenta and scattering angle, and the regions 
of integration for the total cross sections can be found from the figure. 
Since the cross section for neutron-proton scattering, for example, is 
far from simple in the relevant energy regions, only numerical calcula- 
tions have been made. Figure 26 shows the general results to be ex- 
pected: the curves correspond to special choices for the cross section. 











- 






10°/ \ 


- 


45° 


20V'' 


/ V 
/ \ \ 




\^ 









50 
Energy (Mev) 



100 



Fig. 26. The energy distribution of nucleons scattered at a fixed laboratory angle 
by a single collision with a nucleon in the nucleus. The effects of the Fermi motion 
and of the exclusion principle have been included. The incident energy is 100 Mev; 
an empirical fit to the neutron-proton scattering cross section was used. The vertical 
strokes mark the energy expected at the three angles shown if the collision was 
with a free stationary nucleon, in which case the energy would be a unique function 
of angle. The spreading effect of the Fermi motion is conspicuous (65). 

The smearing of the energy-angle correlation and the shifting of the 
scattering toward right-angle scattering is at least made qualitatively 
evident. All this applies of course to an event in which only a single 
primary collision is made within the nucleus. 

2. Following the Cascade: The Monte Carlo Method. The absence 
of any theoretically based analytical form for the nucleon-nucleon 
scattering cross sections, and the rather complicated geometry in coor- 
dinate and momentum space required to treat the problem, have led 
to the use of an interesting method which enables not only calculation 
of the results of a single collision, but also actually follows out step by 
step the development of what may be called a nuclear cascade. 

In this application of a general computational scheme, called for 
obvious reasons the Monte Carlo method, a single incident nucleon of 
given energy is allowed to enter the nuclear volume. Its collisions are 



Sec. 11A] Nuclear Reactions at High Energy 149 

followed in complete kinematical detail, and the subsequent collisions 
of all its collision partners are followed as well until either: (1) any of 
the excited nucleons passes outside of the nuclear volume, after which 
of course no further interest attaches to it except that it is an emitted 
nucleon, with its energy, direction, and charge; or (2) any of the nucleons 
involved, while still within the nuclear sphere, reaches an energy below 
some previously selected value, after which it is regarded as captured, 
and its excitation energy passes eventually into the motion of an excited 
compound nucleus. Then follows the process of nucleon evaporation, 
regarded as separate from the initial cascade. The cascade computations 
are usually made graphically. The distinguishing feature of the method 
is that the decision as to which of a set of equally probable events occurs 
is made by some procedure of random choice, using tables of random 
numbers, spinners, etc. Various simplifications of geometry may be 
made, and the continuous functions of angle, etc., which occur are 
typically stepwise approximated by dividing the interval into a set of 
discrete subintervals, giving the function some constant value as an 
approximation throughout the whole of the subinterval. 

Let us follow a neutron of energy E incident on a nucleus of fixed A 
and radius. Where does it make its first collision? Its point of entry 
is chosen by making a random choice among the rings of equal area 
normal to the direction of the beam. Then its distance of travel is 
found by assuming that it moves with a mean free path given by the 
assumed total interaction cross section with free nucleons of the given 
density (the effect of the exclusion principle can come in only after the 
collision has been made). A distance 5 which is small compared to X, 
the mean free path in nuclear matter, and which gives a probability of 
collision 5/X, can be chosen, and a random decision made as to whether 
or not it collides in this line segment. The nucleon continues to progress 
segment by segment in a straight line, until either it leaves the nuclear 
volume or it makes a collision. Typically at these energies a fair fraction 
of the nucleons traverse the nucleus without collision, especially those 
which enter near the edge. When the nucleon has made a collision, a 
whole set of random choices must be made. First, the momentum of 
the struck nucleon drawn from the Fermi gas must be found, by pick- 
ing at random from intervals of equal probability in momentum space, 
weighting the flat Fermi momentum distribution by the dependence 
of the collision cross section on relative momentum. Then the angle of 
scattering must be found. Usually it is best to work in the center-of- 
mass system, and to approximate the differential cross section for scatter- 
ing by a dozen or so steps. Once the angle of the collision has been found 
by a random choice, the final momentum vector of each of the collision 



150 A Survey of Nuclear Reactions [Pt. VI 

partners is fixed uniquely by the conservation law, since the collision 
is regarded as free. Now one may see if the collision is allowed by the 
exclusion principle. If both the final vector momenta do not lie outside 
the filled sphere, the collision is nullified, and the original particle con- 
tinues undisturbed until it makes another collision. If the collision is 
permitted, then each of the collision partners is now a nucleon of definite 
vector momentum at a specified point in the nuclear sphere. The 
history of each can now be followed until it in turn escapes or is absorbed. 
Account is of course kept of all partners in each collision, and the event 
associated with one incident nucleon is not complete until every struck 
nucleon has been emitted or absorbed. It is obvious that the com- 
putation is tedious; it is equally clear that it is an extremely flexible 
method, capable of high accuracy. It is hardly a calculation so much as 
an experiment in thought, using a well-defined model. 

Two calculations have been worked out on these lines for incident 
energies 90 and 400 Mev, and reported in some detail (BIO, G5). They 
are unfortunately not strictly comparable, mainly because different 
fundamental cross sections for collision were used. In one case the target 
nucleus was lead, whereas in the other it was A = 100, more like the 
heavy nuclei of photographic emulsion. In Fig. 27 is drawn a typical 
cascade event just as it was followed. Here the approximation was made 
of treating the nucleus as a two-dimensional circle, after the first collision; 
the graph then gives directly the projected angles of emission. In the 
event shown a 400-Mev nucleon entered, making its first collision about 
one-third of the way across the nucleus. The two partners of this colli- 
sion were nucleons initially of 431 and 13 Mev of kinetic energy. It 
will be noted that a nuclear potential well of 31 Mev is assumed, with an 
additional barrier of 4 Mev (half the Coulomb barrier, since protons 
and neutrons are not distinguished). One of the first pair of collision 
partners leaves the nucleus directly, as a fast cascade fragment of 282 
Mev. The other goes on to try to make a collision in a short distance, at 
the point marked by a circle, but it is forbidden to do so by the exclusion 
principle. Then it goes on to an allowed collision with a rather fast 
nucleon of the Fermi gas. In all, this event consisted of six nucleon 
collisions: one nucleon enters the nucleus with 400 Mev kinetic energy 
outside; and three leave, taking with them over 300 Mev, mostly in the 
kinetic energy of the one very fast collision partner of the first collision ; 
while four more are excited but "captured," leaving the residual nucleus 
excited by 66 Mev, with the mean binding energy per particle taken at 
about 8 Mev. 

The collection of results for both calculations gives some impression 
of the nature of nucleon cascades. The mean excitation energy of the 



Sec. 11A] Nuclear Reactions at High Energy 151 

nucleus changes very slowly with the energy of the incident particle: 
the 90-Mev neutron on lead left a mean thermal excitation energy of 
some 44 Mev, the 400-Mev proton on emulsion nuclei only about 50 
Mev. This slow rise of excitation with incident energy is partly an 
artifact of the somewhat forced comparison, but it is actually to be 



28 Mev 




7 Mev 



400 Mev 



282 Mev 



Fig. 27. A projected diagram of a nuclear cascade event, followed by the Monte 
Carlo method. The entering nucleon has 400 Mev of kinetic energy; the numbers 
mark the energy of each of the nucleons participating. The path of each nucleon 
is shown until it leaves the nucleus, or until it reaches an energy below 35 Mev, 
taken as the threshold for capture. Along the path of each particle an effective 
collision is marked with an open circle; a cross implies a collision which was for- 
bidden by the exclusion principle. Note that in this event three cascade particles 
emerged, all in the forward hemisphere, and the nucleus was left excited thermally 
with about 65 Mev, enough to evaporate several neutrons isotropically before be- 
coming stable. From (B10). 

expected as a general result, since the high-energy nucleons make high- 
energy secondaries, which typically depart from the nucleus with most 
of their energy, since the mean free path rises with nucleon energy. 

In both calculations also there is a rather wide distribution of energies 
among the cascade products, and a considerable fraction of the energy 
almost always goes to them, as the low mean excitation indicated. 
With the lower energy and the bigger nucleus, however, about 5 percent 
of those incident particles which made any collisions whatever left their 
whole energy in nuclear excitation, with no cascade particles emitted. 
In the high-energy, A = 100, case, no event of this kind occurred at all 



152 A Survey of Nuclear Reactions [Pt. VI 

(among sixty studied); the minimum energy found in the cascade 
particles was about half the incident energy, 200 Mev. 

The general nature of the cascades can be seen from Table 12 and 

TABLE 12 

Nuclear Cascades Studied by the Monte Carlo Method 

90-Mev 400-Mev 

Neutron Proton 

on Pb on A = 100 
Fraction of incident particles traversing 

nucleus without any collision 0.15 . 33 
Fraction of cascades without emerging 

particles 0.05 

Mean thermal excitation energy 43 Mev 50 Mev 
Mean number of cascade particles 

emerging per cascade event 1.2 3.2 
Maximum number of emerging cascade 

particles 3 <~6-8 

Figs. 26 and 27. It should be remembered that the sampling error 
alone in all these calculations is still quite large. Only a few hundred 
cascade particles have been "seen to emerge," and information about 
any special category of them is necessarily only qualitative in character. 

It is quite clear from the results that this process does not much 
resemble the evaporation model. The large number of emitted particles 
with quite high energy, and the strong departure from anything like 
isotropic distribution in angle, are distinguishing features of the nuclear 
cascade. The present experimental material is fully consistent with the 
model we have outlined here. Of course, the evaporation of low-energy 
particles, among them often complex nuclei of A from 2 to 10, with 
isotropic directional distribution, is expected and observed to follow, 
completing the de-excitation of the struck nucleus (Bll, H3). 

In both calculations reported, the neutrons and protons were not 
distinguished, but replaced by a gas of nucleons with about average 
properties. This reflects the feeling that the cascade particles will 
roughly be randomly divided into neutrons and protons, without much 
correlation between particle type and either direction or energy. The 
fact that in heavy nuclei, especially, there are more neutrons than pro- 
tons is perhaps the principal difference to be recalled, apart from the 
collision cross sections. The Coulomb barrier and the binding energy 
differences, especially the symmetry energy, which tends to keep the 
nucleus from becoming either proton- or neutron-rich in excess, will be 
unimportant factors in modifying the cascade, because of the typically 



Sec. 11B] Nuclear Reactions at High Energy 153 

high energies of the emitted nucleons, in strong contrast to the evapora- 
tion mode of disintegration. Some detailed examination of the Monte 
Carlo results tended to verify the idea that nucleons emitted would 
be about half protons and half neutrons, for the nucleus with A = 100 
and a 400-Mev incident proton. It is important to keep this feature 
of the cascade in mind when discussing the experimental data, which 
for the most part give information concerning only the ionizing prongs 
of the star event. There is ample empirical evidence, however, that the 
neutrons are in fact emitted (C8). 

The actual collision cross sections for free collisions are still of course 
not entirely certain. The empirical situation is discussed in Part IV, 
Volume I, but the principal question comes from the very curious ob- 
served behavior of a p . p , which is so strikingly independent of both energy 
and angle above about 100 Mev. Whether this property is one character- 
istic of the special states allowed by the exclusion principle in a proton- 
proton collision, or whether it reflects some deeper difference between 
the behavior of protons and neutrons, is a matter which will much affect 
the easy treatment of nucleons on an equal footing. It is this flatness 
of the cross section which is responsible for the increase of the proton 
energy loss in nuclei of medium size at high energies, and consequent 
growth in star size as the nucleon energy goes from 100 to 400 Mev, 
even in medium- A nuclei. 

B. Correlations among Nucleons 

Of course, the neglect of all nucleon interaction in the Fermi gas model 
is to be regarded as a zeroth approximation. In fact, the nucleons are 
not uniformly distributed either in momentum space or in coordinate 
space. The nuclear matter is lumpy, with constantly forming and dis- 
solving groups of nucleons. When a given nucleon enters into a collision, 
it is not free. Some momentum may be transferred to a neighbor through 
the force binding the two; or the recoil may be regarded as shared by a 
correlated spatial cluster of two or even more nucleons. In the complex 
process of the cascade, with high momentum transfers and repeated 
events, these effects show up only as eventual improvements to our 
present crude theory; but there are already known processes in which 
these correlation effects are, so to speak, the whole story. In the single 
elastic scattering of fast protons by light nuclei (where the cascade has 
no room to develop), in processes of typically low momentum transfer 
(as reactions induced by gamma-rays and by the absorption of slow 
mesons), and in the so-called "pick-up" processes (the inverse of deuteron 
stripping, in which a fast nucleon leaves the nucleus no longer alone but 
now paired with another nucleon which originated in the target, to 



154 A Survey of Nuclear Reactions [Pt. VI 

form a stable fast-moving deuteron), the influence of the correlations 
and their consequent momentum distribution is decisive. 

The probability for finding two nucleons of the Fermi gas at positions 
r x and r 2 respectively is of course given by the integral 

^(ri, r a ) = £ fdr 3 - ■ -dr N \ *( ri - • -r N ) | 2 

ap J 

where the wave function >P is given by the determinantal expression 



^ 



<Pb(ri)<f>b(r2) ■ 



,<Pa(*j) =^e*"-"x(o->) 



Using the orthogonality of the individual plane waves for different 
allowed momenta hk a , and replacing the sum over values of k a by an 

y /•Pmax 

integral over momentum space £ ~~ * ; I dp, we obtain the 

k a (2irh) 3 J 



result 



P(ri,r 2 ) = 22 — 2 (Ufa) 



where the o- li2 give the spin, and t x , 2 the character, of the nucleon at 
points t\ and r 2 . The space function w depends only on the distance 
between the nucleons and is given by 

, | 3(sin 5 — 5 cos 5) . 

w(\ ri - r 2 |) = w(8) = h5 = p max | rj - r 2 | 

o 

(Hlb) 

Thus there is no correlation in position between unlike nucleons, or like 
nucleons differing in spin; but between nucleons alike in type and spin 
there is a definite spatial correlation arising from the exclusion principle, 
even if all forces between nucleons are neglected. Presence of interaction 
forces of course affects the wave function and hence the correlations. 
Using the Fermi gas as a zeroth approximation, calculations have been 
made to find what the spatial correlation is like under various assumed 
forces. Figure 28 shows the general character of the results. The 
momentum space distribution is given with and without interactions. 
The details of course depend on the forces assumed to act, but the gen- 
eral effect is to replace the rectangular Fermi gas momentum distribution 
with one which contains some components of higher momentum — 
traceable in the usual perturbation theory to transitions to unoccupied 
states of the Fermi gas spectrum induced by collisions. The actual 
distribution, with interactions, is somewhat as though the Fermi gas 



Sec. 11B] Nuclear Reactions at High Energy 



155 



were present but not at zero temperature. A calculation by Watanabe, 
assuming certain Gaussian potentials, indicated that the ground state 
of the nucleus of high A might correspond roughly to the momentum 
distribution in a Fermi gas of nucleons at a temperature of 6 or 8 Mev 
(W2). 

There are many processes, mentioned above, which reflect the mo- 
mentum distribution of the nucleons in the nucleus. One of the most 



Fermi gas 




Fig. 28. The momentum distribution of nucleons in a Fermi gas at zero excitation, 
and the same distribution when interactions are considered. Note that the effect 
of interactions in momentum space simulates the presence of thermal excitation. 

interesting is the rather unexpected process called "pick-up." Table 13 
gives a summary of some experimental results which make it plain that 
the pick-up process is by no means a rare one (Y4). 

TABLE 13 
Pick-Up Cross Sections fob 90-Mev Neutbons 
C Cu Pb 



"■inelastic, all events 


0.2 


0.8 


1.8 


barns 


Relative cross section for 










events yielding: 










Cascade protons with 










E p > 20 Mev 


40% 


30% 


25% 




Pick-up deuterons 










{E d > 27 Mev) 


12% 


7% 


4% 





Evidently the escape of the deuteron intact is relatively more difficult 
for the larger nuclei. It is to be expected that a deuteron can leak out of 
a good-sized nucleus at these energies only if it is formed in the far sur- 
face region; otherwise one or the other of the nucleons within it will 
collide in leaving the nucleus. This implies that the formation of such 



156 A Survey of Nuclear Reactions [Pt. VI 

a system is really quite probable when the whole nucleus is taken into 
account. 

The process can be described in rather classical language as the en- 
counter of the incoming nucleon with another nucleon, of the right type 
and moving with the right direction and speed, so that the two move 
along together, forming a deuteron when the binding forces become 
effective. The whole event is by no means localized in a small region of 
space within the nucleus, since the deuteron is a big structure. Pre- 
sumably the nuclear edge contributes most of the pick-up processes, 
when nucleon passages outside the nuclear volume by distances as great 
as the range of the nuclear forces often contribute. Energy and mo- 
mentum must be pretty nearly conserved between the two particles, 
since a strong recoil would imply the break-up of the weak deuteron 
bond. The whole picture is roughly confirmed by experiment; deuterons 
do appear in fair number, coming off mostly forward, with about the 
energy of the incoming particle. At other angles the effect is smaller, 
and the typical energies lower. The incoming nucleon seeks a partner 
moving with the same momentum; then the two could move with 
negligible relative kinetic energy, and deuteron formation would be 
easy. But such a process would rob the nucleus of a great deal of energy, 
since essentially it would then emit two particles each with kinetic 
energy equal to that brought in by only one. The over-all energy and 
momentum conservation then implies emission of somewhat slower 
deuterons. 

The process is treated by an approximation that is physically very 
satisfactory. We make two assumptions: (1) The incoming neutron 
interacts with only a single proton at a time. All multiple events, sub- 
sequent collisions, etc., are neglected, to be treated separately. We will 
find the cross section for a single proton and simply multiply the result 
by Z. (2) During the collision, the forces between the collision partners 
are so strong that they overwhelm all the forces which bind the struck 
nucleon into the nucleus. The whole effect of the binding forces, which 
are by no means neglected, is thought of as determining the initial 
momentum distribution of the struck nucleon in the original nuclear 
ground state. For this second assumption, the approximation has been 
given the name of the impulse approximation; it has found application 
in a variety of problems involving collisions with a complex system (Cl). 

We write the usual form for the cross section per proton from the 
time-dependent perturbation theory: 

2x . .„ 
d*df = — Pf\ H of | 2 dE 
n 



Sec. 1 IB] Nuclear Reactions at High Energy 157 

Call the final deuteron momentum, measured in the center-of-mass 
system of the struck nucleus, hK; the initial neutron momentum hk . 
Write the energy of excitation of the residual nucleus W/ — Ef — E , 
and the deuteron binding energy B. If we normalize the continuum 
states of the initial neutron and the final deuteron center-of-mass motion 
to unit volume, we have for the cross section per proton, for deuteron 
emission into solid angle dfi leaving the nucleus with excitation Wf, 



2tt ,__,„. 2*- M 2MhK , 

C 
1 M 2 K 



^=^1^0 = --.— |F 0/ N0 



27T 2 h 4 k 



■dSl\V of \ 2 (112) 



The matrix element | V f | is that of the neutron-proton interaction 
taken between the initial and final states normalized as described. 
These would be unit-amplitude plane waves for the incident neutron and 
the outgoing deuteron center-of-mass if we used the Born approximation, 
but the present calculation is not restricted to that simplification. For 
present purposes, we will simplify by neglecting the change of deuteron 
momentum with Wf, which is good in the limit of high neutron momen- 
tum, and take merely some average Wf. Now let us take as given the 
undisturbed wave function for the proton in its initial state, bound to 
the nucleus. Call this &(%), writing it as usual as a function of the 
proton position, r p . We can find from this by Fourier transformation the 
momentum-space wave function, 



"^-T^/^,*, 



(2x) 

where now | <pi | 2 is the probability for an initial proton momentum k p . 
The chief assumption of the impulse approximation now tells us that 
this initial momentum distribution alone affects the process, which in all 
other respects behaves as though the proton were initially free. Then 
over-all conservation requires the two conditions : 

(a) K = ko + k 

(113) 

h 2 X 2 M 2 _ 

(b) = B-\ + W f 

W 4M 2M 

We will also write the wave function for the neutron-proton system in the 
initial and the final states as fe, ^ respectively. Here each is a "deu- 
teron" wave function: in the initial state, the "deuteron" is of course 



158 A Survey of Nuclear Reactions [Pt. VI 

highly excited, with a large positive energy, while in the final state the 
outgoing deuteron is bound, in its ground state, \f/ , of internal motion. 
These functions, like the interaction potential, we take to depend only 
on the relative coordinates of neutron and proton; the center-of-mass 
coordinates simply lead to the required momentum conservation. In 
the initial state the relative internal momentum of the neutron and 
proton is given by (k p — k )/2 = K/2 - k . The factor of one-half 
arises in the usual way from the introduction of the reduced mass in the 
equivalent one-body problem expressing the relative motion. Now 
the squared matrix element of the interaction, which determines the 
cross section, can be written, for a single proton, 

| H of | 2 = | w (kp) I 2 X | fdr+ *(r n - x p )V(x n - r p )fo(r re - r p ) 

(114) 

The content of the impulse approximation is just the writing down of 
this expression, first, to be summed over all the protons, and, second, 
as a simple product of the two-particle scattering matrix element with 
the probability for a given proton initial momentum. Now we can write 
the wave functions and scattering potential V(i n — r p ) in the momentum 
representation, giving them as a function of the relative momentum, 
say q, and the matrix element can be written 



(0 V \D) 



^o(q), ^(q)^ 



»(f-kb) 



(115) 



Thus the scattering amplitude in this two-body collision determines the 
whole process, using (114). In principle, this amplitude could be ob- 
tained by a complete knowledge of neutron-proton scattering experi- 
mental values, though the most important range of q is not easily studied 
in this way. The energy conservation (113) can be used to fix | q |. A 
related calculation has been carried out numerically by Heidmann (H5). 
It is instructive to apply the Born approximation to our somewhat 
more general result. For simplicity let us neglect spin questions, and 
consider only the triplet states of the original collision, putting a factor 
of % into the cross-section formula (112). Then we will replace the 
initial neutron-proton wave function fe by a plane wave, giving simply 
a delta-function in momentum space. We can write for V just T n . p — E 
and the matrix element for the scattering, (0 | V | D), becomes 

(B + h 2 (K/2 - k ) 2 /M) X «,(K/2 - k ) 

The operator T — E has been transposed, so that it acts on the final 



Sec. 11C] Nuclear Reactions at High Energy 159 

state wave function <p . With this done, we can write the entire quantity 
| H f | 2 , which we need to sum over all the protons, in the form 

D | H„ | 2 = Z\ w | 2 X (B + h 2 (K/2 - k ) 2 /M) 2 X | * (K/2 - k ) | 2 

V 

(116) 

The first factor Z\ </>i\ 2 is simply the number of protons to be found with 
initial momentum K — k in the original nucleus; the second factor is 
the energy operator; and the third factor just the probability for a 
definite relative momentum q = K/2 — k in the deuteron ground state. 
With a reasonable choice of \f/ , say the familiar Hultheh (C2) approxi- 
mation, iAo(r) = const. (e~ ar — e~^ T )/r, it turns out that the last two 
factors roughly compensate, so that the variation of as with neutron en- 
ergy is dominated almost wholly by the proton momentum distribution. 

The cross sections given by the impulse approximation certainly 
account for the rather surprising frequency of pick-up deuterons, and 
for their distribution in energy and angle. Indeed, the calculations 
apparently overestimate the frequency of pick-up, presumably because 
subsequent interactions break up the nascent deuterons. But it is most 
satisfactory that this process, so unexpected at high energy, can, espe- 
cially in just that region, be given such a simple explanation in terms 
of momentum transfer. 

The pick-up process is the first of many found to lend themselves to 
the impulse treatment, and hence found to depend very much upon the 
momentum distribution within a nucleus : elastic scattering of nucleons, 
photon absorption, and many meson processes, like production by 
nucleons and gammas and capture by nuclei, are in this category (C3, 
F15). The description of the nuclear matter in momentum space, both 
by statistical and by more detailed methods, is sure to become more 
important and more familiar, supplementing the present description 
given most frequently in coordinate space only. 

C. The Optical Model for the Scattering of Nucleons 

At high energies, as indeed even at lower ones, the simplest of scatter- 
ing experiments is the measurement of the over-all attenuation of a 
beam of incident nucleons. If the experiment is done under conditions 
in which the energy is well-defined, and with so-called "good geometry" 
(i.e., an arrangement such that particles which have deviated even 
slightly from the original beam direction are excluded from detection), 
it is possible to measure also the fraction of the beam which has been 
elastically scattered. The convenience of such experiments has led to 
the use of a quite abstract and general nuclear model, which is tailor- 



160 A Survey of Nuclear Reactions [Pt. VI 

made to give elastic and total cross sections without at all worrying 
about the details of the nuclear collisions. In an earlier section we saw 
how the assumption that the interaction between neutron or proton and 
nucleus was strong led to the idea of a sticky nucleus, opaque to the 
passage of nucleons, and then to a total cross section which for energies 
sufficiently high became just twice the geometrical cross section of the 
nucleus. The elastic scattering was represented by the portion of the 
emergent wave which produced the ordinary shadow behind the nuclear 
obstacle by interference with the incident wave. For a black obstacle 
we saw that this gave a coherent elastic, or "shadow scattering," cross 
section which was just equal to the inelastic cross section %R 2 . Its 
angular distribution was also determined. The optical model here used 
(F6) is a straightforward generalization of the results obtained earlier, 
extended to the case where the obstacle is no longer black but "gray," 
for the free path for nucleon collision in nuclear matter is no longer 
negligible, as we have seen above, in the hundred-Mev region. The 
ideas of physical optics, however, in the familiar Kirchhoff-Fraunhofer 
approximation, become better and better because %/R <3C 1. 

For this model, we think of the nucleus as a gray refracting sphere. 
Its opacity, or optical density, and its index of refraction for the Schrod- 
inger waves characterize the model. Then the amplitude of the scattered 
waves, and the damping due to absorption, can be computed, generally 
neglecting all interface reflections, and considering only "volume" 
effects, in analogy with the W.K.B. approximation. In its simplest 
form, the model regards all nuclei as uniform gray spheres, which differ 
only in diameter. This geometry is of course too simple, and in no way 
required by the optical approximation. It may be better to think of 
the nuclear matter more realistically as thinning out gradually toward 
the edge from a uniform central core; there is some evidence for such a 
model in exactly this sort of calculation. 

The nuclear matter is assigned a complex refractive index, N = n + it. 
Here n is the ratio of the nucleon wave number k at a given point within 
the nuclear volume to the wave number k Q = (2ME) 1A /h of the incident 
nucleon outside in free space (neglecting Coulomb forces). The quan- 
tity r specifies the amplitude damping within nuclear matter; the wave 
amplitude changes by a factor 1/e in a traversal of a distance within the 
nuclear matter equal to one wavelength of the incident particle, \(/(x) 
= io-e~ xk ° T , and is related to the mean free path for interaction in 
nuclear matter I — y^k^r. The inelastic scattering which we will calcu- 
late includes all exchange and absorption effects; it depends only upon 
the imaginary part of N, that is, on t. The real part of the index of 
refraction will determine the coherent scattering, in which the scattered 



Sec. 11C] Nuclear Reactions at High Energy 



161 



wave represents particles of the same type as those of the incident beam 
and having exactly the same energy in the center-of-mass system. We 
will consider only the uniform-sphere model; for this, N is constant 
within the nuclear sphere of radius R, and unity everywhere outside. 

In the shadow of the sphere, the transmitted wave which passes at a 
distance from the central diameter of the sphere (see Fig. 29) emerges 
with an amplitude a(p)e' koR , where e lk ° B is the amplitude of any portion 
of the incident plane wave which misses the sphere. The region of the 
shadow must contain a wave of lessened amplitude whose intensity 



Beam 




Source plane 



(a) 



(6) 



Image 
plane 



Fig. 29. The geometry of the optical model for nuclear scattering. In (a) the beam 
is shown incident on the nuclear sphere of radius R. The path length of a particular 
ray will clearly be 2s. In (b) the geometry of the diffraction calculation is shown. 
The source and image planes are normal to the incident beam direction. For each 
scattering angle 6 and azimuth * in the image plane, the total contribution is found 
by integrating the contributions of every radius and azimuth angle 4> in the source 
plane. This is the familiar Fraunhofer procedure. 

decrease gives the total incoherent absorption cross section, simply the 
total contribution to 1 — | a | 2 integrated over the whole area of the 
obstacle. To obtain the coherent, elastic, or diffraction scattering, one 
has instead to find the net amplitude which, added to the incident plane 
wave in the shadow region, will yield the transmitted amplitude a. This 
is plainly 1 — a, and gives a cross section | 1 — a | 2 . 

In the case of a spherical obstacle, the factor a(p) is evidently 



(j( n \ iko(B — 2s) (n+ir)2ikos Jko(n — 1) -2s — 2rkos 


(117) 


with s 2 = R 2 — p 2 . Integrating 1 — | a | 2 , we obtain the absorption 


cross section: 


tfabe = I Pdpi dp(l - e- iTSk ") = 2,r I s ds(l - e- 2 " 1 ) 
Jo Jo J 


[ 1 - (1 + 2R/l)e- 2R ' l l 
" ' B L 1 2RVP J 


(118) 



162 A Survey of Nuclear Reactions [Pt. VI 

The elastic scattering is rather more complicated. It is given by the 
integral (F6) : 

•rdiff =fpd P f *d<p\ 1 - e -*<v<-2^-i)*o> p (n9) 

The angular distribution of the diffraction scattering can be obtained 
in a similar way. The approximation here is to sum the contributions 
reaching a given image point, far behind the sphere, from every source 
point in a plane wave front of amplitude a — 1 within the geometrical 
shadow. Then the Kirchhoff integral (of optics) can be written 



const 
R 



ffd<p p dp e ik ° R - pA B| X[1- a(s)] (120) 



where the geometrical relations are those of Fig. 29. Using polar co- 
ordinates in the effective source and image planes, we can write 

R-v 

i r = p(cos <p cos $ + sin <p sin $) X sin @ 

and the entire integral becomes 

"p dp d<p e ik ° 8in e cos ( *-*V(s) - 1] 



If 



Using a familiar representation for the Bessel function, we can write (F6) 

p dp J (k p sin @)[1 — a(s)] (121a) 



-J 

•'o 



and the differential scattering cross section dc/dil is just proportional to 
I «ac | 2 - We can evaluate the constant in the limiting case of R/l —+ <». 
For a totally black sphere, this goes over to the case already given in 
Eq. (46), namely: da/dti = R^J^koR sin 6)/sin @] 2 . For the gray 
spheres here under consideration, the evaluation can be carried out. 
By replacing the integral over p by the sum over I, in which I + ^ = kp 
and the relation Jq[(1 + 2) s * n ®1 = Pi( cos ©)> valid for large I and small 
@, is employed, we find 

l + V 2 <kB 

a sc ~ E (21 + l)Pi(cos »)[1 - a(si)] with 

si = [(k R) 2 -(1 + \fV A IK (121b) 

This is very reminiscent of the partial-wave analysis of scattering, and 
indeed is just the W.K.B. approximation to the phase shifts of the 
Rayleigh partial wave procedure. That procedure has been used ex- 



Sec. 11C] Nuclear Reactions at High Energy 



163 



actly (PI), giving a result somewhat larger than does (121b) at small 
angles of scattering. A series procedure is also applicable for the evalua- 
tion of (121a) directly. 

The results of such a calculation are indicated in Fig. 30. The effect 
of nuclear transparency is to reduce the contribution of the shadow 
scattering as the transparency increases (or R/l decreases). The graphs 
show this reduction in the case of a plausible value for the index of re- 
fraction, implying some increase in phase velocity within the nucleus. 



2.0 - 



03 
J= 1.0 



^ 


M=1.16 


"totA^ 2 

<W*--R 2 


°abs/Tfl 2 



1.0 



R/l 



2.0 



Fig. 30. Cross sections computed from the optical model. The diffraction, absorp- 
tion, and total cross sections, measured in units of the geometrical cross section, 
are given as a function of the nuclear radius measured in units of the free path. A 
particular value, n = 1.16, has been chosen for the refractive index. From (F5). 

When n is not simply unity, the diffraction scattering may considerably 
exceed the geometrical limit wR 2 for a black nucleus. 

By fitting the data on neutron scattering at 90 Mev to the results of 
Eqs. (118) and (119), we can choose n + ir in such a way that the value 
of R obtained for a certain choice of n + ir fits for all nuclei from Li to U 
the simple relation R = r A H , with r chosen at 1.39 X 10 -13 cm, a 
very reasonable value. Suitable values, which are consistent with direct 
scattering experiments for neutron-proton and proton-proton scattering 
at this energy are I = 3.3 X 10 -13 cm and n = 1.16 (F5). For these 
values the coherent scattering is larger than the geometrical value. 

A less empirical evaluation of n + ir is possible in two ways. The 
nuclear material simply acts to change the phase velocity of the waves, 
and can thus be represented by a potential well of depth V, with the 
relation n = k/k = (1 + V/E) y% . This works well enough for 90-Mev 
neutrons, but the same choice by no means satisfies the experiments at 
higher energy, where the variation of total cross section with energy 
does not follow the predictions of the simple potential well idea. A 
much more sophisticated approach has been made (J2) by means of the 



164 



A Survey of Nuclear Reactions 



[Pt. VI 



familiar result of optical dispersion theory, which gives the refractive 
index in terms of the scattering amplitudes for forward scattering by the 
individual nucleons. The relation is 

n - 1 = K-„(0) + a n . p (0)]Trp/k 2 (122) 

where p is the density of nucleons in the nucleus, and the a's are the 
appropriate scattering amplitudes. Use of the empirical cross sections 



da/dil 
barns 




24° 16° 8° 0° 

Neutron scattering angle 

Fig. 31. Angular distributions of elastic scattering computed from the optical 

model. The differential cross section is shown as a function of scattering angle for 

two nuclear radii (corresponding to copper and to lead). The experimental points 

are shown. All this is for 90-Mev neutrons incident. From (F5) and (CIO). 

is required for I, .but it is of course not enough to fix the quantities a in 
(122) unambiguously. Best fit with the observed total cross sections for 
a nucleus like Al is found when the real part of the index of refraction is 
made to go rapidly to unity, k/k — > 1 — as the energy increases above 
100 Mev. Such a scattering amplitude results from a nucleon-nucleon 
potential consisting of a strong repulsive core surrounded by an attrac- 
tive well, with a core radius of some 10 -13 cm. This result can be taken 
as some confirmation of the view, based on the very flat and strongly 
isotropic proton-proton cross section above 100 Mev, that some such 
"hard-core" model of the nucleon-nucleon interaction is at least phe- 
nomenologically suitable. With a n . v = a p . p the fit for the total cross 
section to relation (119) is quite good. 

The angular distributions are another source of information. In Fig. 
31 we show a comparison between theory and experiment for neutrons 



Sec. 11D] Nuclear Reactions at High Energy 165 

of about 90 Mev. The diffraction pattern is qualitatively correct, but 
the experimental peaks are a little sharper, as though the nucleus were 
somewhat smaller than expected from the constant-density model. 
Experiments at higher energy show the positions of the successive 
minima in the diffraction pattern, and indicate that the minima are in 
fact blurred out, perhaps only slightly in the heaviest nuclei, but beyond 
recognition in the lightest ones. This is consistent with the results to 
be expected if we surrender the naive idea of a strictly uniform sphere of 
nuclear matter and imagine instead that the nucleus has a core of con- 
stant density surrounded by a fringe of nuclear matter with a thickness 
something like the range of nuclear forces within which the density 
falls gradually to zero. Such a model is of course actually implied by 
the simple notion of a nuclear surface, as it is for any classical drop of 
liquid. Taking such a model, and the connection between n and the 
empirical nucleon-nucleon cross sections given by the dispersion theory, 
Jastrow (J3) claims a reasonable fit, both to the total cross sections and 
to the details of the angular distribution of the diffraction scattering, 
over a wide range of energy and nuclear mass. Much more remains to 
be done. 

D. Processes of Nuclear De-excitation at High Energy 

Let us return to the topic first mentioned in this chapter: the detailed 
course of nuclear disintegration in the region of hundred-Mev energies. 
In the step-by-step treatment of the cascade, each particle was followed 
until the kinetic energy it possessed, relative to the bottom of the over- 
all nuclear potential well, fell below a certain limit, some 30 Mev. After 
this, no single collision between this particle and the nuclear edge is 
likely to result in the emission of the particle; rather, reflection will 
occur, and the energy will gradually distribute over the whole of the 
nuclear system. Not until it is again by chance concentrated on a single 
nucleon (or in a single well-defined mode of nuclear motion) will emission 
occur. This is the basis of the compound nucleus treatment which is the 
main burden of this entire discussion. To place matters in terms of the 
familiar thermal analogy: excitation of a nucleus by collision with a 
single nucleon, or a small nucleus, corresponds to heating the nuclear 
matter very hot indeed in a small spatial region. From this region a 
nucleon or several may emerge, taking off most of the energy. But 
the heat will proceed to spread. The time of conduction of energy over 
the whole, compared with the time it takes a single nucleon or a few 
nucleons of a cascade to emerge, will determine whether or not the subse- 
quent steps will proceed by thermal evaporation. Our treatment 
assumes that, after a certain energy loss has been suffered by each fast- 



166 



A Survey of Nuclear Reactions 



[Pt. VI 



recoiling nucleon, all the remaining excitation energy passes into thermal 
form. Since the spread of energy is mainly through the zero-point 
motion of the Fermi gas of nucleons, this implies that the statistical 
treatment will be valuable when the individual nucleon has an excitation 
not large compared to its zero-point energy, well under 20 Mev or so. 
This limits the process plainly to cases where the whole nucleus is excited 
by a good deal less than its total binding energy, say by some 5A Mev at 
the most. We have already seen that such excitations are ample to 
handle the most frequent cases observed in the bombardments with 



Bi(p, 3n) 




10 20 

Energy of incident protons (Mev) 

Fig. 32. Reaction cross sections for various proton-induced reactions on bismuth. 

The competition between the several modes of decay is shown strikingly here; as 

each new mode becomes energetically possible, it tends to drain away the previously 

favored decay method. After (K5). 

nucleons up to almost 500 Mev in fairly heavy targets, and they cover 
also, at least in not too small nuclei, the interesting phenomena following 
upon meson absorption, both w and n (B12, M6). 

The statistical treatment we have used earlier in the low-energy cases 
implies of course a thermal equilibrium, which will not actually exist 
in the case of high excitation and the emission of many particles. We 
can expect that a kind of transient equilibrium will in most cases exist 
between the successive nucleon emissions which make up the total event. 
The limit on such an idea is no different from that mentioned just above: 
the time for rediffusion of the excitation energy after an act of emission, 
governed by the thermal transport, should be small compared to the 
delay before the next concentration of energy on a single emitted particle. 
The typical thermal transport time is under 10 -21 sec; while the time for 
nucleon emission up to the highest excitations here considered is ten 
thousand times as great. The whole division of the process into an 
immediate cascade and a subsequent evaporation is of course approxi- 



Sec. 11D] Nuclear Reactions at High Energy 167 

mate; but within these limits the approximation is successful to a rather 
high degree, and very instructive. 

Of course, the transition from the ordinary statistical treatment of 
reactions like (n,a) or (n,p) to the large evaporative stars so dramatically 
revealed in the nuclear emulsion studies is a continuous one. For in- 
stance, an elegant series of experiments by Kelly (K6) has carried the 
simple competition theory of formation of the compound nucleus and 
its successive decay (see Section 7) through the whole series of reactions 
with the target nucleus Bi 209 : (p,n), (p,2ri), (p,3n), (p,4n). Excitation 
energies in this experiment ranged up to about 40 Mev. Figure 32 shows 
the results obtained, which fit the expected relations of the statistical 
theory of competition [see Eq. (89)] very well indeed. If we write the 
ratio of excess energy of excitation over threshold to the temperature 
as x, the data fit the simple result a{p,2n)/a v = 1 — (1 + x)e~ x very 
well by adjusting the single parameter, the temperature kT. The value 
chosen for kT agrees completely with the calculated value for the semi- 
empirical level density formula, Eq. (31), about 1.1 Mev. 

1. Spallation and Evaporation Stars. As the bombarding, and hence 
in general the excitation energy, continues to rise, we reach a region of 
higher complexity. Here not only a single product, or a few related 
competitors, can emerge, but a whole series of successive steps can be 
taken, each one in many alternative ways. This fact, and the often 
important fluctuations away from the mean statistical behavior of even 
such a complex multiple evaporation, is the reason for the absence of 
any well-codified study of this general subject. So far we can give fairly 
satisfactory but rather tedious means of calculation, but no compact 
summary of the possible results. 

Two different experimental approaches to the field have led to different 
nomenclature, and to somewhat different descriptions, of one and the 
same phenomenon. The radiochemical procedures of separating the 
various radioactive products from targets bombarded with projectiles 
at differing energies, and so obtaining yield curves for a whole series of 
possible products, is one powerful means of study (B4). Here of course 
only the over-all change is observed; if A and Z change by 20 and 10 
say, the event can be described as the emission of 10 neutrons and 
10 protons. But the emission of alphas or even other small nuclear 
fragments in such an event is by no means excluded. Typically a high- 
energy bombardment yields a bewilderingly large number of product 
nuclear species. By extension from the idea of the fission of a nucleus 
into two large fragments, with a little neutron spray, the radiochemists 
have come to call these processes spallation, with the implication of the 



168 A Survey of Nuclear Reactions [Pt. VI 

emission of a whole series of small fragments, rather than the cleavage 
into two big ones familiar in fission. 

On the other hand, the cosmic-ray workers, first with their cloud cham- 
bers (P4), and more recently with the powerful nuclear emulsion tech- 
niques (B9, H3), have typically studied not the statistical residue of 
millions of disintegrations by identification of the product nuclei, but 
the ionizing fragments released from each individual nuclear break-up, 
with the familiar bristle of ionized rays, each marking the trajectory of a 
single ionizing fragment starting in the central decaying nucleus and 
ending at the end of the fragment's range. In this method, besides the 
simple count of the fragments and charge carried off, the energy and 
angular distribution can also be obtained. But a good statistical sample 
is tedious to collect, and neutrons are invariably missed. (A few ex- 
periments with counters have satisfied us that the invisible neutrons, 
expected in more or less equal numbers with the visible "prongs," are 
really emitted.) The typical appearance of the event in the emulsion 
has led to its designation as a "nuclear star." Star or spallation reaction, 
the event is the same, but the method of detection, and to some extent 
the features used for description, differ widely. 

In Fig. 33 we present a rather generalized summary of the results of a 
typical spallation study: the bombardment of a copper target with 
protons of 340 Mev (B4). The contours in the N,Z plane show the ob- 
served yields, with some plausible extrapolations and smoothings. 
Characteristic is the large number of observed nuclear species, more 
than thirty having been identified. The biggest yield is for nuclei 
differing by only a few units in Z and N, plainly due to events in which 
not much excitation was left behind after a small cascade. There is a 
strong tendency to stay in the vicinity of the stable valley; it is not 
likely that the neutron-proton balance will be badly upset by evaporative 
events in which nearly all the emitted nucleons are protons (or neutrons). 
This tendency is compelled by the quasi-equilibrium theory, and favored 
by the energetically advantageous emission of alpha-particles themselves. 

The spallation studies have shown one other rather interesting phe- 
nomenon (B5). Both by identification of the short-lived isotope Li 8 , 
and by arguments of energy conservation, it has been demonstrated 
that at least a few events are possible which bridge the gap between 
simple alpha-emission and straightforward fission. In Section 10, it 
was pointed out that the splitting-off of small fragments is energetically 
favorable for a wide range of A because of the balance of Coulomb and 
surface energy. In spallation experiments, where infrequent events 
can be detected with relative ease, it has been found that there is indeed 
a small yield of nuclei of mass far removed from that of the target even 



Sec. 11D] Nuclear Reactions at High Energy 



169 



for rather low bombardment energies. For example, 70-Mev protons 
on Cu give a small yield of CI 38 . The connecting reaction which in- 
volves emission of single nucleons and alphas only is the reaction 
Cu 63 (p,6apn)Cl 38 . But that reaction has a threshold, calculated from 
mass differences, of some 110 Mev. The extreme reaction, from the 



Sq25 - 




Fig. 33. A contour map plotting the yield of a spallation reaction in the N,Z plane. 
The target nuclei are marked with solid circles; copper was bombarded with 340- 
Mev protons (B5). The yields center roughly on the line of greatest stability, shown 

dashed. 

point of view of minimizing surface energy, is the three-particle break-up, 
Cu 63 (p,nAl 25 )Cl 38 . The threshold for this rather unlikely course of 
reaction is about 50 Mev. It thus seems established that a variety of 
heavier fragments, of which Li 8 has surely been found, but probably 
extending over a wide range of masses, up to processes splitting the 
nucleus into several pieces of approximately equal weight, can in fact 
take place. This marks a kind of transition to the case of low-energy 
fission, and implies that the statistical competition of specialized vibra- 
tions of the nucleus leading to massive splits will have to be taken into 
account in a complete theory of cooling. 



170 A Survey of Nuclear Reactions [Pt. VI 

2. Evaporative Stars: The Fundamental Process. The study of 
the energy and angular distribution of observed star prongs enables a 
rough distinction — allowing considerable overlap — between the cascade 
and its evaporative consequences. The evaporative prongs are gen- 
erally marked by: (1) a roughly isotropic angular distribution, (2) a 
rapid decrease in prong frequency as energy increases. The long tail 
up to high energies actually observed — so-called "gray" and "sparse 
black" prongs, named from their low ionization densities in the emulsion 
(B9) — are regarded as cascade particles. True evaporative particles 
ought to show a typically quasi-Maxwellian drop-off of number with 
energy, and no such long tail to energies far beyond any possible mean 
particle energy. In general, we expect the evaporative process to be 
sensitive, especially in its last stages, to details of nuclear structure, like 
the odd-even property and even more refined shell-like behavior. 

The thermodynamic treatment of the stars is based on the statistical 
formula, due to Weisskopf, which connects the probability for dis- 
integration of a compound nucleus with the formation cross section, by 
the principle of detailed balance. [Compare the discussion in Section 5, 
Eqs. (51) and (52).] We write 

gM p(f) 

2 h 3 p(i) 

where P(T) dT is the probability per unit time for emission of a particle 
of mass M and spin statistical weight g with kinetic energy in dT; a is 
the cross section for capture of the same particle by the residual nucleus / 
to form the initial nucleus i; and the functions p are the level densities 
of initial and final nucleus as indicated, functions of course of the mass 
and charge number and of the initial excitation energy, W, and final 
excitation W. 

In principle, we may now use the function Pj(T) for emission of a 
particle of type j with mass A 3 - and charge Z } - from the initial nucleus of 
given kind and given excitation energy, W = W + T + Bj, with B } - 
the binding energy, and then simply follow the course of the successive 
emissions until all the excitation energy has been converted into kinetic 
and potential energy of the fragments, and into radiation. This implies 
the solution of the set of coupled equations which can be given, if only 
formally, as (F13) 

dy(Z,A,W,t) „_ r x 

' = E dW'y(Z + Z h A + AjWdPjiW'.W' - W) 

dt j JW+Bj 



P(T)dT = ^—- ff (T)T — dT (123) 



- Z y(Z,A,W,t) f P 3 {W,T) dT (124a) 



Sec. 1 ID] Nuclear Reactions at High Energy 171 

Here the yield function y(Z,A,W,t) gives the probability of finding a 
nucleus with charge Z, mass A, excitation energy W, at time t after the 
start of the process, which fixes the initial condition: 

y(Z,A,W,t) = Saa, 5zz„ S(W - Wo) (124b) 

Now from the function y one can learn all the facts about the star, 
including of course both the mean behavior and fluctuations from it. 
For example, the total yield of a definite nuclear species is just given by 

Vtot(4,Z) = f 'y(Z,A,W,t -» oo) dW 

Naturally such a complete treatment is next to impossible; indeed, 
fluctuations are generally treated very casually [but see (Tl)]. We 
shall examine a more manageable approximation; perhaps the complete 
treatment will one day be performed by Monte Carlo technique, which 
seems very well suited to the task. 

3. The Mean Behavior in Cooling. The most satisfactory approach 
is the study of the mean behavior of the evaporative process. To make 
this possible in general (F13), several approximations must be made, 
not all of easily controlled accuracy: 

1. In the fundamental formula (123), the cross sections are replaced 
by very simple forms: usually just the geometrical irR 2 for neutron emis- 
sion, and a simplified expression like <r = wR 2 (l — V'/T) for a charged 
particle with a Coulomb barrier height given by V = 0.7(ZZje 2 /R), to 
take care of penetration effects. 

2. The level density is expressed by some one of the results of a par- 
ticular nuclear model, discussed in Section 3. The detail with which the 
shell properties are represented may vary greatly. 

3. The cooling proceeds usually by the emission of particles whose 
Aj and Zj are small compared to those of the initial and residual nucleus. 
This allows series expansion of functions of A and Z, performed as 
though A and Z were continuous variables, instead of integers. 

First of all we define the relative width (hence probability) for emitting 
a particle of a given kind, indexed by j. In the typical calculation, it 
may be worth while to consider half-a-dozen or more types of emitted 
particle, as proton, neutron, deuteron, H 3 or triton, He 3 , He 4 or alpha- 
particle, and sometimes even heavier fragments. In general, then, the 
mean emission width is given by the expression 

Tj(W) = f Pj(W,T) dT 

JVf+Bj 

where Pj is the transition rate given by (123). The approximation of 



172 A Survey of Nuclear Reactions [Pt. VI 

mean cooling is of course just to consider the process as going always 
proportional to these mean emission rates, and to neglect the different 
courses of the evaporation which come from the fluctuations about the 
mean energy and type of decay. Naturally no single act follows the 
average behavior, and the successive steps in fact depend in turn upon 
just what particle and energy loss preceded them. These refinements 
are here neglected. Now we can calculate the mean energy of excitation 
lost by evaporation of particle j: 



X 



Pj(T)TdT 



8Wj = Tj + Bj = + Bi (125a) 



X 



PjdT 



and we can express the mean cooling in one way by writing the mean 
rate of energy loss per nucleon emitted : 

— Z IV HWj 
SW j 

■=- = (125b) 

8A Sr,- 5Aj 

In a similar way we can write the mean number of particles of type j 
lost per unit energy loss : 

Srij Tj 

= = = (125c) 

8W HTjdWj 

and for the entire cooling from initial to final excitation energy get 

— r w * Tj{W) dW 

dnj = | ^ (125d) 



■ _ r w < Tj{W) dW 

' 3 Jvj'+bj ]r Tj s~Wj 



Here there is of course one such equation for each of the values of j, say 
six or more, and the equations are all coupled. The main coupling is 
through the effect of the changing neutron-proton ratio, and some 
reasonably trustworthy solutions have been obtained (neglecting other 
relations) (L3). 

(a) Cooling Behavior with an Explicit Model. To gain any insight 
into the expected behavior of this phenomenon, it is necessary to intro- 
duce an explicit nuclear model which can fix the level density and its 
variation with Z,A, excitation energy, and even finer details like the 
odd-even effects. In Section 3 we set out a number of expressions from 
various models for the level density p(W). We choose here to use the 
rather simple model of the Fermi gas, which has some weak experimental 



Sec. 11D] Nuclear Reactions at High Energy 173 

support, and can be regarded as the empirical expression of the results 
for nuclei of middle weight and for energies in the range here studied. 
With a Fermi energy of 22 Mev, we obtain [compare Eq. (31)] 

^(.A.Z.W) (AW)** 

P (W) = v and S = 0.63 (126) 

P ^ ; t(2t dW/dr)* Mev 

Here the entropy is given as a function of the excitation energy W, and 
we have introduced the temperature t, writing W/Mev = At 2 /10, and 
t is in Mev. With this model the rate of evaporation becomes, neglecting 
some slowly varying ratios, 

P(T) dT = gM/ir 2 h 3 (T - V')e^~ Si) dT (127) 

and, if we make the approximation, valid for small emitted fragments 
and 8W « W,, of expanding the function S(A,W), we get S(A — SAj, 

W - 8W) = S A (W) - SW/t - SAjt/10 and 

~(T - V) 
P{T) dT = const (T - V) exp (128a) 

T 

This is the familiar Maxwellian distribution of evaporated fragment 
energies. The mean kinetic energy taken off by an evaporated particle 
i is 

I x 2 e x dx 
Tj = t — = 2r (128b) 



3 ,00 

xe~ x dx 



I 



and the total width T 3 {W) for emission of j-particles with any energy 
T is approximately 

Tj = h P(t,W) dW 

JBj + Y' 

~ a — i ■ exp I 1 t in Mev (129) 

~ y 2x 11 Mev V 10 t / 

Both terms in the exponent here depend strongly on the type of 
particle emitted; they are the effective potential barrier V/ and the 
binding energy Bj. Now we must explicitly give the variation of 
binding energy with A and Z, which will of course determine the nature 
of the most probable emitted particle. We write a semi-empirical form 
for the binding energy of a nucleus in its ground state [compare Eq. 



174 A Survey of Nuclear Reactions [Pt. VI 

(101)], with <(JV - Z)/A ) written for the value of (N - Z)/A which 
corresponds to the most stable nucleus of a given A : 

B -= const + Cl A - c 2 A[(N - Z)/A - { (N - Z)/A >] 2 

Here we can take c x about 8.6 Mev, and c 2 about 23 Mev. These num- 
bers are suited for target nuclei in the middle of the table, A ~ 100, as 
in photographic emulsion. Now, if a particle of type j is evaporated, 
with 8n,j neutrons and SZj protons, taking off kinetic energy T, the drop 
in excitation is given by differentiating B, and regrouping the terms, 
to yield 

8W = T + Bj 

[N - Z IN - Z\l 
Bj = Cl dAj - 2 |— \~-J~ / J < c " bn * ~ c * SZ ^ + 7 > 

f a IN - Z\ l\N-Z IN - Z\l) 

c " =C2 { 1 - A M\-i-/-i[-x- + \-x-)JH a8c2 

f a IN - Z\ 1VN-Z IN - z\-\) 

(130) 

and Ij is the internal binding energy of the emitted fragment. The 
energy Bj is of course the threshold energy for emission of the given 
fragment with zero kinetic energy, which we call the binding energy of 
the fragment into the initial nucleus. We have considered only nuclei 
rather near the stable valley — (N — Z)/A ~ {{N — Z)/A ) — in evalu- 
ating c n and c z . 

Now we can examine at least the general cooling behavior by simply 
looking at the competition between the various emitted fragments. 

Q>) The Competition in Cooling. Let us look at the ratios of the r,- 
for the various particles. First, note that the statistical and mass factors 
g 5Aj alone will give relative weights for the six most common products 
as follows: p:n:d:T:He 3 :a: :1:1:3:3:3:2. Note the increase in statis- 
tical probability, especially for the heavy isotopes of hydrogen, which 
feel a low Coulomb barrier as well. 

Much more important, however, than the statistical and mass factors 
is the exponent in Eq. (129). We write here the expected emission width 
relative to that for neutron emission, using the energy relations of (130) . 
We can write the effective Coulomb barrier heights for nuclei of the 
emulsion (A ~ 90) as V/ = 6bj Mev, where the values of bj are esti- 
mated to include charge, radius, and mass effects on penetrability, and 



Sec. 11D] Nuclear Reactions at High Energy 175 

we get b n = 0; b p = 0.7; b^ = 0.8 = br; &He 3 = 1-6. Then the widths, 
relative to the width for neutron emission, become 

r„/r„ = l 

iyr„ = exp [- (92(i. -v) + 4)/r] 

T d /V n = 3 exp [-t/10 - (56(i- - v) + ll)/r] 

T T /T n = 3 exp [-2t/10 - (18^ - v) + 14) /r] 

r H ea/r n = 3 exp [-2t/10 - (110(? - r) + 19)/t] 

TjT n = 2 exp [-3t/10 - (74(* - p) + 8)/r] 

The energies involved and the temperature r are all given in Mev. We 
have written v for the quantity (N — Z)/A ; the value of v for the most 
stable nucleus of a given A is written as V. Thus {v — v)A = neutron 
excess — proton excess. 

An examination of these ratios alone enables a qualitative discussion 
of the course of the mean curve of cooling. The most noteworthy 
features are these: 

1. The Coulomb barrier — contributing to the last term of the ex- 
ponent — of course favors the emission of neutrons over charged particles, 
and reduces the emission of particles of charge +2, as long as it is of 
any consequence. Since the temperature r is ~(W)**, the effect of the 
Coulomb barrier on charged-particle emission will be decisive for all 
excitations up to some critical energy. For higher energies the emission 
of ions and neutrons will show little difference ascribable to Coulomb 
effects. The critical energy for proton barrier effect in these middle 
nuclei is an excitation of from 100 to 150 Mev; for helium isotopes, 
from 250 to 350 Mev. 

2. Apart from the Coulomb barrier, the most striking term in the 
exponent of the relative emission widths is the symmetry energy, propor- 
tional to the fractional neutron excess of the initial nucleus, (v — v). 
Looking, for example, at the proton-neutron ratio, T p /T n , we see that 
this term favors neutron emission when there are too many neutrons 
compared to the region of stability, and favors proton emission when 
the nucleus is neutron deficient. This term has been called a "governor" 
term (L3), because it prevents great excursions from the region of stable 
nuclei in the cooling process. It arises of course from the parabolic 
form of the familiar Heisenberg valley in the nuclear energy surface. 
In many-particle stars it will more or less insure that more neutrons 
than protons are emitted. 

3. The leading term in the exponent of the relative widths for emission 
of the complex star fragments — d, a, etc. — grows more negative as the 



176 



A Survey of Nuclear Reactions 



[Pt. VI 



excitation, and with it the temperature, increases. This term arises 
from the dependence, not of the energy, but of the entropy itself, on the 
number of nucleons in the nucleus. It favors the emission of lighter 
rather than heavier particles, because more ways exist of assembling 
a nucleus if it contains more nucleons. If it were not for this type of 
term, the emission of the heavier fragments would become more prob- 
able than that of single nucleons because of the purely statistical weights. 




Fig. 34. The course of the neutron number of an evaporating nucleus as a function 
of nuclear temperature. The neutron excess is plotted; it is the difference between 
the number of neutrons in the nucleus, N, and the number of neutrons characteristic 
of the most stable nucleus of the same mass number A. The neutron excess first 
increases and then decreases as the nucleus cools. The terminal value of N — (iV) 
is about —1, implying the subsequent emission of a slow proton, or simply /3 + -decay, 
following the main process of evaporation. See pages 174-177. The curves are 
drawn for a nucleus with initial A = 100, and for two initial temperatures, 4 and 8 

Mev. After (L3). 

For example, without this term r a /T n — > 2 as t grows without limit. 
As star size increased, then, alpha-emission would outweigh proton 
emission and even neutron emission, a very odd result, and far from what 
is actually observed. With this entropy term, however, the ratio 
Ta/Tn goes through a very flat maximum in the region of 100-Mev 
excitation. As we have seen, however, in earlier chapters, the variation 
of level density with A is quite poorly known, and detailed predictions 
from this simple entropy assumption cannot be expected to agree in 
detail with experience. 

Now we can integrate the simultaneous equations (125) to give the full 
course of the cooling. This has been done in rather rough approxima- 
tions, perhaps best by LeCouteur, whose procedure we follow (L3). 



Sec. 11D] Nuclear Reactions at High Energy 



177 



In general, we can expect this typical state of affairs : the highly excited 
nucleus, say with 400 Mev, begins by evaporating neutrons and protons 
about equally. To begin, it has v — v almost zero — perhaps one or two 
protons and neutrons have been knocked out of the stable target nucleus 
in the nuclear cascade. But now the nucleus boils off several protons 




200 400 600 

Initial excitation energy (Mev) 

Fig. 35. The number of evaporated particles of several types as a function of initial 
excitation energy, for an evaporating nucleus of initial A = 100. The neutron num- 
ber scale is at the left; that for charged-particle emission at the right. After (L3). 

and several neutrons. By the time it has emitted say four of each, the 
cooling nucleus has become neutron-rich, with respect to the smoothed- 
over line of maximum stability, and the "governor" term begins to 
favor the loss of neutrons. Then the cooling proceeds, somewhat prefer- 
entially losing neutrons, until the excitation has dropped below some 
100 Mev. From here on, proton loss will be almost prohibited by virtue 
of the Coulomb barrier, and the excitation energy will fall until no more 
particles can be emitted. The resultant nucleus is now neutron-deficient 
by a little less than one neutron on the average, more or less independent 
of the initial energy of excitation for not too low excitations. We show 
in Fig. 34 a schematic plot of the neutron number against excitation 
during the cooling, which follows the course here described. 



178 



A Survey of Nuclear Reactions 



[Pt. VI 



In Fig. 35, we plot, after LeCouteur, the mean numbers of various 
emitted particles for different excitation energies, with an initial nucleus 
of A — 100. The virtual independence of the relative yields of various 
particles upon energy is a feature of the theory; but the calculations have 
actually been done neglecting the small variations actually predicted, 




10 20 30 

Kinetic energy of evaporated particle (Mev) 

Fig. 36. Energy spectrum of evaporated particles for several initial excitation 
energies. Initial nucleus of A = 100. The solid curves all refer to emission of pro- 
tons; the dotted curve, to that of alpha-particles. After (L3). 

and using the relative yields given by equations for an initial excitation 
of 400 Mev. In Fig. 36 we give the expected kinetic energy distribution 
of evaporated fragments; it is given by the rough averaged formula, 
very much as in (127), by 



P(T) AT = 



V 



-(T-V)/t 



dT 



(132) 



Results are given for several excitation energies, and for both singly 
charged and doubly charged star prongs. 



Sec. 11D] Nuclear Reactions at High Energy 179 

All these results are likely to be acceptable only for fairly large excita- 
tions (W > 150 Mev), and the theory cannot be taken seriously for A 
much less than about 60. The actual figures are for a definite A = 100. 

(c) The End of the Cooling Process: Low-Energy Prongs. We followed 
the cooling process down to the point where the temperature is so low 
that no further nucleon emission is possible energetically. But this 
threshold of course varies with the type of particle. Moreover, the 
process has left us with a nucleus which is neutron-poor by about 0.8 
neutron. Then the values of the threshold energy are not the same as 
for a stable nucleus, and become, from (130), B p c^7 Mev, B n ~ 10 Mev, 
5 a ~ 5 Mev for the three main possibilities. Thus, even though the 
Coulomb barrier reduces the rate of charged-particle emission for such 
low energies very much indeed, the neutron cannot compete at all, 
since the binding energy of the neutron in the neutron-poor residual 
nucleus is higher than normal. De-excitation can go on by gamma- 
emission, beta-decay, or charged-particle emission. The anticipated 
lifetime for these slow processes at an excitation of 10 Mev is perhaps 
10~ 16 sec or so for the gamma, and some 10 -2 sector the beta. Protons 
emitted through the barrier with kinetic energies of anything above 
0.5 Mev will favorably compete with such slow radiative processes. 

Alphas and other fragments of mass above one unit will experience a 
much greater loss in penetrability; since this factor is already 10 -10 , 
protons are about the only practical competitors. Thus, if a nuclear 
particle of any kind, the next-to-the-last to be emitted, leaves the residual 
nucleus with an excitation of less than 10 Mev and of more than B p + 0.5 
or some 7 Mev, a proton will be emitted. If the remaining excitation is 
below B p , gamma-rays alone can come off. Thus about 30 percent of all 
nuclei left with 10 Mev excitation will emit slow protons, far more than 
would be expected if neutron competition were not excluded by the shift 
in binding energies due to the neutron deficit. A computation (L3) has 
shown that there is to be expected in emulsion nuclei about 0.2 proton 
per star with kinetic energies in the range from 0.5 to ~3 Mev. These 
very slow protons have apparently been observed (H3). In addition 
there is a considerable excess of slow alphas observed; this cannot be 
accounted for by such a mechanism, but is ascribed to a lowering of the 
effective barrier by processes to be considered in Section HE (L3). 

It will not escape the reader that this type of calculation has over- 
looked even the most striking features of nuclear shell structure, like 
the odd-even variation in energy content. This does not affect the 
previous phenomenon, slow-proton emission, but in general it should 
have some detailed effects on the closing scenes of the evaporation, when 
excitations are not large compared with the few Mev which represent 



180 A Survey of Nuclear Reactions [Pt. VI 

shell-closing energies, etc. Moreover, the rather large gamma-ray width 
which seems to be observed in the neighborhood of 20 Mev excitation 
may occasionally compete — say one time in ten or a hundred — with 
the last one or two emitted particles. All these interesting but somewhat 
fine points await both more systematic experiment and a better theoreti- 
cal treatment. 

(d) Effects of High Nuclear Temperature. The calculations of evapora- 
tion ought not to neglect the changes in the nuclear properties which 
arise from the high nuclear temperatures during the earlier stages of the 
process. In equation (126) and in the binding-energy formulas of (130) 
we have over-simply considered the nuclei throughout as though they 
were in their ground state. The work of LeCouteur (see Figs. 34 and 
35 which are essentially taken from his work) actually did not leave 
out this important and complicated circumstance. The effects to be 
considered are three in number, all closely related to thermal expansion: 

1. Depression of the Coulomb barrier, and consequent favoring of 
emission of low-energy charged particles, as a result of the thermal 
expansion of the nucleus. An estimate of the thermal expansion of the 
nucleus due to an actual change in the equilibrium volume energy at 
high temperature, entirely analogous to the familiar phenomenon on 
large scale, can be obtained by thermodynamic arguments. We can 
write the internal energy of a nucleus of given A as U(V,t), showing its 
dependence on temperature and volume. For a definite temperature — 
actually t = — this internal energy is a minimum at some volume, say 
V . Expanding, we can write 



dU 

U(r,V) = U(t,V ) + — 






(V-V ) 2 +- 



But the coefficient of the linear term is zero by the definition of V - 
Now the Helmholtz free energy, for negligible external pressure, is just 



F = U - rS = U(r,V ) - tS(t,V) + - 



(v - v y 



ld 2 U 
2~dV 2 

and the actual equilibrium volume at any temperature r will satisfy 
= fixing the free energy at a minimum. We have 



dF 

the relation — ■ 

dV 



already given the entropy in terms of A (Eq. 126) and we can estimate 
the entropy as a function of volume by using the relation V = ^irr 3 -A. 
This of course is not precise; the entropy depends both on A explicitly, 
the number of nucleons, and on the volume for fixed A, but it will serve 



Sec. 11D] Nuclear Reactions at High Energy 181 

for an estimate. Now we get S = 0.2At = 0.2 • F/(4Trr 3 /3) • t, and this 
yields an expression for the volume change with temperature: 

V = V + [0.2/(W/3)] • [r 2 /(d 2 U/dV 2 \ TjVo )] (133) 

The familiar thermodynamic result dF = —S dr — p dV for small depar- 
tures from equilibrium yields the relation dF/dF| T = —p. But the ordi- 
nary isothermal compressibility is defined as k = —(1/V) (dV/dp)\ T . 
Since dp/dV\ T = l/(dV/dp)\ T , we have l/« = Vd 2 F/dV 2 \ T . For low 
temperatures we can write, closely enough, d 2 F/3V 2 | T = d 2 U/dV 2 \ T . 

An estimate of the nuclear compressibility, d 2 U/dV 2 , is easy to 
obtain in several ways. An application of the virial theorem (78) 
gives d 2 U/dV 2 = fc£7 kin /F 2 , where the factor k is about 2; direct calcu- 
lations with models using a Fermi gas and correcting for assumed two- 
body interaction potentials are in fair accord. Some insight may be 
had by recalling that the velocity of sound v is given by the relation 
w 2 = -[V(d 2 U/dV 2 )]/p. If the sound velocity is estimated very 
roughly as simply the rms velocity of the particles in the Fermi gas, we 
have v 2 /c 2 = 6E F /5Mc 2 . Using (133), we can write the relative change 
in radius upon heating [R(t) - R(0)]/R(0) ^ %(r 2 /E F ) = 0.008t 2 
(t in Mev), a small but not entirely negligible change. Bagge (Bl) 
has pointed out that there is likely to be another, more dynamical, 
source of reduction of the Coulomb barrier than this uniform volume 
expansion. Surface vibrations in the excited nuclear drop "wrinkle" 
the surface; over many cycles, protons in the nuclear surface layers are 
to be found farther out from the center than in the smooth undisturbed 
drop at zero temperature. The diagram of Fig. 37 shows the nuclear 



a 
Q 



n~\ 



OR OR OR 

(a) (b) (c) 

Fig. 37. Schematic plot of density of nuclear matter in a cross-sectional cut through 
a nucleus. In (a) is shown the abstract model of a uniform sphere without surface 
effects; in (b), the more realistic version of the Fermi model, with a fringe of grad- 
ually falling nucleon density; in (c), the nuclear density in a [highly excited and hence 
much expanded and strongly oscillating nucleus. 

density distribution at zero excitation and at high temperature; there 
is both a lowering of the mean p and a wide fringe of nuclear matter as t 
increases. An estimate of a 2 , the mean square amplitude of the surface 
vibration normal modes at a given temperature, has been made by Bagge, 
who shows that although the number of modes excited is not large, and 



182 A Survey of Nuclear Reactions [Pt. VI 

quantum treatment is necessary, the value of a 2 is nevertheless very 
closely proportional simply to the nuclear temperature. This can be 
described by saying that the surface vibration energy is simply a definite 
fraction of the total excitation (F14) and that the mean frequency of 
the modes excited changes slowly. The surface effect appears to be quite 

important, reducing the barrier in the ratio V'/V — 1/(1 + "^ a 2 /R), 
and a semi-empirical estimate of the effective proton barrier as V p ' 
= 4/(1 + 0.15r) seems to fit the present data reasonably well. With 
this variation of barrier with temperature, the spectrum of emitted 
charged particles of course changes, and this reduction of V is fairly 
surely observed (Bll, B8). 

2. The excited nucleus has a different balance between protons and 
neutrons; at high excitation there is a tendency toward increasing the 
difference of neutron and proton number, N — Z. This of course tends 
to favor the emission of protons from a nucleus already rich in neutrons. 
The tendency originates from a complex set of causes, most important of 
which is a new balance between potential and kinetic energy which has 
to be struck for the now expanded volume of the heated nucleus. A 
shift in volume decreases the Fermi kinetic energy, and the potential 
energy will readjust to fix a new minimum total energy, now at a different 
value of v from that of the cold nucleus of the same A. The effect has 
been evaluated by LeCouteur most recently, using a Fermi gas plus 
interactions. He shows that it leads to an entropy term of the form 
+ const [(N — Z)/A] 2 . It becomes appreciable only for excitations 
above about 300 Mev. It has only a rather small influence therefore. 
It more or less corrects the high-energy emission for the fact that stable 
heavy nuclei in the ground state are neutron-rich, and equalizes the 
proton-neutron emission at high excitation. 

3. The surface tension energy at high temperature decreases. Evi- 
dently the nuclear expansion could reach a "critical point" where the 
nucleons are all excited beyond their binding energies; here the drop 
would vaporize entirely. The nuclear surface tension will decrease 
regularly with temperature until the critical temperature, t c , is reached. 
Using the rough Fermi-gas-like relation between excitation and tem- 
perature, we can write W c = 8A Mev = At c 2 /10, and from this we get 
t c = 9 Mev, independent of A in this approximation. Now let us again 
take thermodynamics as a guide (G7). If we write down the Gibbs 
free energy for a surface film, in which the surface tension —7, surface 
force per unit length in the film, takes the place of pressure in the usual 
case, we have Gs — U — tS — yA for an area of film A, with internal 
energy U and entropy S at temperature r. Now the partial derivative 
— 8Gs/dA\ T = y gives the surface tension. An isothermal change in 



Sec. 11D] Nuclear Reactions at High Energy 



183 



film area will obey the first and second laws: dU = rdS + y dA, and 
we can write dGs = —Sdr — A dy. Differentiating, — dy/dr = (S/A) 
+ (dGs/dr). But, in equilibrium at any r, the Gibbs free energy of film 
and bulk liquid are equal for a given mass of nuclear matter. We write 
G s = G L . Then -dy/dr = (S/A) + (l/A)(dG L /dr) = (S - S L /A). 
Now we can form Gs/A = Gl/A = — y + (U/A) — (rS/A), and we 
get a differential expression for the surface tension 7 as a function of r : 



-T + 



dy _ (U- Ul) 
dr A 



(134) 



The term pV in the liquid-phase free energy is cancelled by that of the 
surface film, except for thickness changes, which we neglect as usual. 



Mev 




9 

Temperature (Mev) 

Fig. 38. The nuclear surface tension as a function of nuclear temperature. The 
curves indicate different models: curve (a) arises from the actual surface energy of 
a classical liquid drop with surface waves excited by the thermal motion; curve (b) 
from the simple linear assumption for the integration of the free energy equation 
of the text. The two are very close. From (Yl). 

Now the right-hand side of (134) is the additional energy due to the 
surface film, measured per unit area of nuclear surface. Using as unit 
of area just 4tt 2 , this becomes about 14 Mev for nuclei in their ground 
states, with r = 0, as the semi-empirical formula showed in Section 10. 
As the temperature rises, surface vibrations are excited and the energy 
increases. Using a liquid drop model, Bethe (B13) has shown that the 
excess surface energy increases like r A /y(r). As a first approximation 
we can neglect the change in y, and write U — Ul/A = 14 + const t a . 
We can then solve (134), using as boundary condition the requirement 
that y(t c ) = 0. The surface tension then goes to zero, following the law 
7(7") = 14 + t/t c — 15(t/t c Y a . A numerical integration has been per- 



184 A Survey of Nuclear Reactions [Pt. VI 

formed by Yamaguchi (Yl) which gives a very similar result. A plot 
is given in Fig. 38. 

The reduced surface tension means that the balance between Coulomb 
and surface energy is displaced in favor of the Coulomb repulsion. We 
would expect this, since the Coulomb forces show only a slow change 
with mean separation, while the surface forces are, of course, of very 
short range. An expanded nucleus requires little additional energy 
to undergo fission, for which process the repulsive Coulomb forces over- 
come the surface tension; the nuclear volume energy can be taken as 
remaining constant throughout. The familiar Bohr-Wheeler estimate 
of fission threshold, as given in Eq. (107), is just 



E f = iTr 2 A 2A -y(r)f 



Q 



With this model, Fujimoto and Yamaguchi (F12) have estimated that 
the silver fission threshold would fall from some 50 Mev for the ground 
state of the nucleus to half of that value at about 100-Mev excitation, 
and to only a few Mev at 350-Mev excitation. The fission width can be 
estimated in the usual way as about 1/ ~ (r/27r)e~ E//T . This makes 
such fission a good competitor for really large excitation, 300 to 400 Mev 
or more. Such heavy spallation fragments are found not infrequently, 
as we have mentioned above (H3) . Of course, fission yielding fragments 
with much smaller mass is even more favored by the decrease in surface 
tension, and many reasonably heavy fragments have been seen in high- 
energy stars, especially the easily recognized unstable ones like Li 8 . 

The mechanism for fission here described is only a single one which 
may contribute to fission at high excitation energies. With the heaviest 
nuclei, and at somewhat lower energies, fission is observed subsequent 
to the emission of many neutrons, at temperatures below the barrier 
height for protons. This upsets the Coulomb-surface energy balance 
as well, and reduces the fission threshold. The mechanism is plainly 
very different (G4). 

It has been observed (F14) that a calculation of the temperature 
dependence of the surface energy, using the model for surface vibrations 
which Bagge employed, yields not very different results, and still predicts 
very easy loss of highly charged fragments at high temperature. Using 
Bagge's formulas for the mean amplitude of surface waves, but taking 
into account the variation of surface tension with temperature, it appears 
that excitations of 300 or 400 Mev would greatly distort the surface 
shape of nuclei, so much indeed that whole little drops would come off 
in a kind of spray, (a 2 ) l/% /R ^ 1 describes such a condition. It seems 



Sec. 11 D] Nuclear Reactions at High Energy 185 

rather likely that the actual situation is more nearly a non-equilibrium 
local heating, capable of distorting the surface locally enough to pinch 
off a small drop, but not involving the surface of the entire nucleus 
in such a way as to give the nearly flat distribution of fission frag- 
ment masses which the near-disappearance of the equilibrium surface 
tension might lead one to expect. The whole phenomenon is clearly 
of importance, but our present account is too closely classical to trust 
quantitatively. 

4. Fluctuations in Cooling. We have so far described in detail only 
the average behavior of the cooling nucleus; we have ignored all fluc- 
tuations. Obviously, such a complicated process in which only a few, 
or perhaps a few dozen, particles take part must be subject to important 
fluctuations in the number, type, energy, and direction of emission of 
the emitted nucleons. So far this problem is only sketchily understood, 
but a simple model will serve us at least for orientation (F12). 

Consider a nucleus so very hot that we can ignore its cooling, even 
though it has emitted many nucleons. It is restricted, moreover, 
to the emission of only one type of particle, with binding energy B, 
which also remains constant throughout the process. Then the prob- 
ability of emitting a single nucleon in energy range dT is just P(T) dT 
— e — T l rr r dT/r 2 , where t is the temperature, taken as constant through- 
out, and T the kinetic energy. Now the probability that the nucleus 
will emit n or more nucleons, when it has an initial total excitation W, 
is given by the multiple integral: 

P(n,W) =J ■ -jdT, ■ ■ -dT n _ l P{T i )P{T 2 ) ■ ■ -PiT^) 

(135) 

n— 1 

< Z Ti < W - nB 
l 

where the upper limit follows because there must remain at least B Mev 
of excitation after the loss of n — 1 nucleons if at least one more is to be 
emitted, and the lower limit represents the process occurring by the 
possible route in which each of the first n — 1 nucleons just manages to 
dribble out with zero kinetic energy. 

It will not seriously distort this rather unrealistic picture to replace 
the Maxwell distribution in kinetic energy by a Gaussian distribution 
adjusted to have a mean and a standard deviation equal to the Max- 
wellian values. We know that T = 2t and (T — T) 2 = 2t 2 , so that we 
can rewrite _ 

, - ( T - ~T) V*r 2 d\T — T) 

2(tt) 3 



P(T) dT = e -< r -*W \ t H ' (136) 



186 A Survey of Nuclear Reactions [Pt. VI 

Now the probability for the multiple emission is a compounding of nor- 
mal distribution. As in the familiar statistical problem of the chi-square 

«-i _ 

test, we write x 2 = 53 (Ti — T) 2 /4t 2 and recall that x 2 is then the 

1 

square of the radius vector in a hyperspace of n — 1 dimensions. In 
such a space the volume element is proportional to the quantity x" -2 dx- 
The proportionality constant is of course a function of n. If we recall 
that, as the excitation W grows without limit P(n, W) — > 1 for any given 
value of n, we can evaluate this function of n. Now for not too small 
values of (W — nB)/r and of n, the region of integration, which is 
bounded by the coordinate axes and a set of planes in the hyperspace, 
can be replaced without serious error of a sphere of radius say Xm in the 
polar coordinate system described by x- Remembering the required 
normalization, we obtain 

/Xmax /*'max 

X n ~ 2 e~ x d x =f{n)\ t in -*>' a e-* dt (137) 

which we can write P(n,W) = y[(n - l)./2, * max (W)]/r[(ra - l)/2], 
where y(n,x) is the incomplete gamma-function defined by 

y(n,x) = j e~H n ~ l dt and Y{n) = y(n, #-»«>) 



It is important to note that we want the dependence of P(n,W) not on 
the value of W, as in the usual statistical problem, but on the parameter 
n. We can estimate the distribution in x 2 > for any fixed large value of n, 
by the familiar saddle-point method, getting a Gaussian distribution 
given by e^* 2 -^- 1 )] 2 /^-!) 2 dy ? n ow t jj e fi rs t approximation to P(n) 
can be obtained by considering the geometrical interpretation of the 
integral, and observing that P(n) is unity for any n such that the sharp 
peak in x 2 lies within the original region of integration. Taking the 
original limits, then, and using the hyperplane rather than the sphere 
boundary, we can estimate that x 2 (H r ,w) = (W — nB)/2r. In the 
next approximation, the width of the region in which P(ri) falls from 
unity to zero can be estimated from the standard deviation in x 2 - We 
are led to these results for the mean number of nucleons emitted in the 
process and for the standard deviation, with an approximately Gaussian 

distribution in n: 

W + 2r 

ll S 

2r + B 

(138) 
-= „ 8t 2 W+2t 

n ■ — n ~ 

B 2 2r + B 



Nuclear Reactions at High Energy 



187 



Sec. 11D] 

The relative rms fluctuation is just An/n ~ 3/(n) M - (r/B). The mean 
energy loss for the emission of a single Maxwellian particle is B + It, 
and the many-particle case is seen to give a mean value not very different 
from what one would get neglecting any correlation, namely W/(2r + B). 
The standard deviation is also not much different from the consequence 
of a very naive statistical estimate. Some reactions have been observed 
in which a highly excited nucleus emits a considerable but definite 



a -a 



O «3 

■§ ° 

PL, 



0.3 


y — 


^* *v 


W- . t . , = 100 Mev 

initial 


0.2 


_ / / 
/ / 








/ / 




X^V 




/ / 




N. \ 


0.1 


1 







2 4 6 

Number of evaporated particles (w) 

Fig. 39. The probability of emission of exactly n evaporation particles as a function 
of n. These fluctuations are calculated, on a very much simplified model, for a 
nucleus of initial A = 100 and excitation energy of 100 Mev. The dashed curve 
comes from an attempt to integrate the coupled equations of evaporation; the solid 
curve, which is a Poisson distribution, from a still simpler theory. Only qualitative 
conclusions may be safely drawn from the present theory. From (Tl) and (H10). 

number of identical nucleons, and the energy spectrum for those emitted 
particles seems to resemble the statistical energy distribution P(n,W) 
given above. 1 

The omission of all the complicating effects from this picture is all too 
clear. A couple of attempts to solve the cascade equations under more 
realistic assumptions have been made (H10, Tl). Even these more 
elaborate efforts have neglected the change in mass and in binding energy 
as the cooling proceeds, and have in general neglected the fluctuation in 
energy taken off with each nucleon. The resulting fluctuations primarily 
emphasize the choice the cooling nucleus makes between protons and 
neutrons, with differing mean probabilities and mean energy removal 
for the two types of nucleons emitted. The results indicate something 
very close to a Poisson distribution in the number of neutrons or in the 
number of protons emitted, with the mean being given by the mean 
cooling behavior we have already considered. In Fig. 39 we plot some 
consequences of the two methods mentioned. It is fairly clear that most 

1 Private communication from E. Segre. 



188 A Survey of Nuclear Reactions [Pt. VI 

of the details of the process have been suppressed in these generalized 
models. Whether the fluctuations are underestimated or overestimated 
by these models is not clear. The changes in mass number and energy 
which have been neglected certainly cause correlations to be more im- 
portant, but some of the correlations have the effect of stabilizing the 
process closer to the mean. It is very likely here also that a Monte 
Carlo procedure will be the first to yield a reliable answer to the problem. 

E. Mesons : Virtual and Real 

We have come to the end of a sufficiently long, but yet only very sum- 
mary, account of the theory of nuclear reactions in general. It is ap- 
propriate to close with an admonition. 

All our considerations have been based on a single picture of the 
nucleus: physically, as a collection of neutrons and protons with more or 
less strong interaction; formally, as a system whose Hamiltonian was a 
sum of kinetic energies of a given number of heavy nucleons plus more 
or less complicated interaction potential energies. It is clear that this 
idea, taken either physically or formally, is at best an incomplete model. 
Looking at the nuclear system in short time intervals, or with fine 
distance resolution, we would not in fact distinguish an unchanging 
number of nucleons colliding and shifting position, but fixed in number 
and type. On the contrary, we know that even the exchange of mo- 
mentum which is the ordinary force between nucleons, like the exchange 
of spin and of charge which correspond to more subtle but still phe- 
nomologically described interactions is not to be understood in this way. 
The mediation of some kind of field of mesons seems certain. Like all 
fields whose quanta need relativistic description, that of the mesons 
within the nucleus cannot be assigned a definite particle count. We have 
to think of transient states, in which mesons appear and disappear, 
carrying momentum, charge, and spin back and forth among the heavy 
nucleons in what may be a very complicated manner. In a way we have 
used a description of the nucleus whose molecular analogue would be a 
pair of protons and an oxygen nucleus held together by some given 
potential, to form a molecule of water. Yet the meson field is still more 
complicated than that of the atomic electrons, though it contains charges. 
It is in some ways more closely akin to the essentially relativistic electro- 
magnetic field, in the cases where the photons may have a purely tran- 
sient existence, as ordinary photons do in fact have within the near-zone 
field of a radiating dipole. The anomalous nucleon magnetic moments 
arise from such meson fields. The existence of many-body forces, of ve- 
locity-dependent forces, of circulating currents not ascribable simply to 
orbital motion or to spin nipping of the nucleons is the very least to be 



Sec. HE] Nuclear Reactions at High Energy 189 

expected from the virtual mesons within the nucleus (SI). Suspected 
are severe changes in the familiar properties of free nucleons themselves 
due to their immersion in the virtual meson sea (M8) . Transient changes 
in charge — producing doubly charged protons or negative neutrons — 
changes in spin, even in rest mass, are all probable circumstances of the 
nuclear matter, looked at sharply enough. And, when energy enough 
becomes available from outside to satisfy the demands of permanent 
meson liberation, all sorts of new effects can be seen. Meson absorption, 
emission, and scattering, where the mesons fly free of the nuclear region 
to reach our counters or the emulsion grains, are but the most obvious 
effects. All have been studied, and are in active development. What of 
the release of mesons, strongly interacting with nucleons as they do, 
which never leave the nucleus, but are reabsorbed within it to transfer 
energy, momentum, spin, and charge throughout the nuclear drop? Al- 
ready it seems that the sharp rise of the deuteron photo-cross-section 
(B6) in the energy region near and just above the threshold for free meson 
production involves some such explanations. 

All this is far from our simple picture of a closed and tight cluster of 
Z protons and N neutrons. But it is a picture closer to the truth. It is 
well to end this account, then, with the clear warning that application 
of the simpler ideas is reliable at best only in the domain where such 
long time averages are involved that the transient mesons can usually 
be replaced by the smooth forces of our picture, and for energies well 
below the energy at which real mesons can be released, even internally. 
With that warning not forgotten, the theory of nuclear reactions can 
yield results valuable for innumerable applications, for the detailed 
understanding of nuclear structure in a kind of chemist's approximation. 
One day this theory may serve for the construction of a better and super- 
seding picture, based on deeper knowledge of the fundamental nature of 
the nucleon itself. 

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Pt. VI] 



Appendix I 



193 



APPENDIX I 

In this appendix we have collected graphs which permit the collection of such 
cross sections as are plotted in text Figs. 12 through 14. These are mainly graphs 
auxiliary to the calculation of penetrabilities. Complete definitions and theory are 
found in text references (C5, K7, Y5). 



5.0 






ffiff0.15fm.t_ 








4.0 






Up lpjflfjg 








'Tl'(?)'i$$$f 














3 






HC I 06||$1 






"V ( rr} 








2.0 






IHo'B 

^Po°7| 


BoM 


M0.9lllHlllf 


3 


1.0 











0.2 



0.4 

X=E/E barri( . r 



0.6 



0.8 



Fig. A-l. Function for computing the penetrability of the Coulomb barrier without 

angular momentum, Po, for 1 = 0. The penetrability Po is defined as: Pq = e~ 2c ° 

with Co = <7y(x). y(x) is here plotted. Figures 12d and 13a of the text give barrier 

heights for various nuclei, g is plotted in Figs. A-2a and A-2b. After (B13). 



194 



A Survey of Nuclear Reactions 



[Pt. VI 




03 



.a 



-a 



03 






Pt. VI] 



Appendix I 



195 




03 

a, 

"3 



J3 






Ml 



196 



A Survey of Nuclear Reactions 



[Pt. VI 



to 

I 



1.0 






























1 1 


























































X=Vh 






Vi 






































































V2 < 






























































































































































































































































































































1 




































































1.0 








































































































































































































































































































































































































































































































1.5 






























































































































































































o.c 


















































































































s = 2.0j 





















































































































































































































































































































































































































































































































1.0 



y=l(l + l)/g* 



2.0 



3.0 



Fig. 



A-3. The 

-2ClXx.v) 



penetrabilities for 



angular momenta I ^ are defined as 
Pi = e _zo " x ' ! ". The plot gives (Ci — Co)/?, where g is the characteristic orbital 
momentum plotted in Figs. A-2a and A-2b as a function of two parameters: energy 
in terms of the barrier height x and parameter y = 1(1 + l)/<7 2 . After (K7). 



Ok 






















































































































































































































































































































































































































20 
















































































































































& 
























































** 






















































s*% 






















































& 
























































X 






















































"1 


€ 






















































































































































15 


























































































































































































































































































































































11 

























































20 30 40 50 60 70 80 90 

Z 

















































































































































X 


sA-U 














































t w 


c\e 


6 












































rvW^ 


»; 


V, 




















































































































, F 




















































'~*2£i 














































•pvojjii-" 




































































































































































Pr 


ot< 


)I\E 


*=0.15_ 









































































































































































































































































































































































.20 



30 



40 



50 



60 



70 



80 



90 



Kg. A-4. Values of orbital angular momentum l c , such that the contribution to 
the total cross section for compound nucleus formation, a = ttX 2 2 (21 + l)Pi, 

1-0 

for all I greater than l c , is less than 10 percent. These curves form a guide for stopping 

the computation of the Pi as I grows. 

197 



198 
z 

(atomic 
number) 



100- 
90- 
80- 
70- 
60- 

50- 



30— 
25 

20— 



A Survey of Nuclear Reactions 



[Pt. VI 



10— 
9 
8 
7 

6— 
5 



(a) 



E 

in Mev 
(center of. 
mass) 
30- 



20— 
15- 



10- 
9- 
8- 
7 

6- 
5- 



2 
1.5- 



1.0- 
.9- 
.8- 
.7- 
.6- 
.5- 

.4— 



.1— 



X 



V 

and 
d 

1.0- 



.7- 
.6- 

.5- 



-1.0 



— .7 

— .6 



(a') 



.1— 

.09- 



- .2 

(a") 



10" 
10" 



'2 10" 

■I io-« 

5 

2 10" s 

4 
6 

8 io- 2 

2 
4 
6 

8 .i 

.2 
.3 

Hb) 

.5 
.6 



- .7 



- .9 



- .95 



a p 
.2 



.09— 

.1 



.15- 



.2— 



.8- 



— .4 



.09 
.10 



-.15 
-.2 



- .7 



(c) 



-.95 



Fig. A-o. Nomogram constructed for the evaluation of the penetrability of the Coulomb barrier, in the 
case of zero orbital angular momentum, I = 0. The penetrability Po can be found in two ways, of 
different accuracy: (1) Using lines (a), (a'), and (a"), the value of the parameter x, which is the energy 
of the particle measured in terms of the barrier height for a definite Z, can be found directly. The 
approximation A = 2Z is used for this graph, which is quite rough for the heavy elements, and may lead 
to Po in error by a factor of ~S, even where Po ~ 0.1. With this value of x, connecting points on lines 
(a), (6), and (c) then gives the penetrability, P . (2) The barrier height B may be found, not from the 
nomogram, but by directly reading off its value from the graphs of Figs. 12c and 13c of the text. Then 
x = E/E, can be found, and connecting proper points on (a), (6), and (c) of the nomogram will give Po 
to good accuracy over the whole readable range of the nomogram. 



Pt. VI] Appendix II 199 



APPENDIX II 

We list first a number of theoretical and experimental survey papers of both a 
general and a more specialized kind. 

Compilations of Experimental Information 

M. S. Livingston and H. A. Bethe, Revs.. Modern Phys., 9, 245 (1937). List and 
discussion of all work on reactions up to July, 1937. 

J. Mattauch and S. Fliigge, Nuclear Physics Tables, New York, Interscience, 1946. 
Some data and references for all reactions up to about 1941. 

W. Hornyak, T. Lauritsen, P. Morrison, and W. A. Fowler, Revs. Modern Phys., 
22, 291 (1951). A very full survey up to the middle of 1950 dealing with reac- 
tions involving the lightest target nuclei only, from H to Ne. 

Books 

S. Devons, The Excited Slates of Nuclei, Cambridge University Press, 1949. Excel- 
lent account of experimental methods and of theory. 

V. F. Weisskopf, in Lecture Series in Nuclear Physics, MDDC-1175, U. S. Govern- 
ment Printing Office, 1947, pp. 106 et seq. Theory. 

J. Blatt and V. Weisskopf, Theoretical Nuclear Physics, John Wiley & Sons, New 
York, 1952. Comprehensive account. 

Papers of More Restricted Scope 

(a) Photonuclear Reactions 

Experimental 

K. Strauch, Phys. Rev., 81, 973 (1951). Work with ~300-Mev y's. 

A. K. Mann and J. Halpern, Phys. Rev., 82, 733 (1951). 
L. Katz and A. S. Penfold, Phys. Rev., 81, 815 (1951). 

Theoretical 

J. S. Levinger and H. A. Bethe, Phys. Rev., 78, 115 (1950). 
E. P. Courant, Phys. Rev., 82, 703 (1951). 

(b) Theory of Particle Reactions in General 

V. F. Weisskopf, Helv. Phys. Acta, 23, 187 (1950). Physical picture. 

E. P. Wigner, Am. J. Phys., 17, 99 (1949). Introduction to the general dis- 
persion theory. 

E. P. Wigner and L. Eisenbud, Phys. Rev., 72, 29 (1947). Presents the full 
theory. 

J. Jackson, Phys. Rev., 83, 301 (1951). A simplification of the theory. 

T. Teichmann, Phys. Rev., 77, 506 (1950). 

E. Wigner, Phys. Rev., 73, 1002 (1948). 

Thelast two present instructive and useful applications of Wigner s gen- 
eral theory. 

(c) Neutron Excitation 

Theoretical 

E. Fermi and L. Marshall, Phys. Rev., 71, 666 (1947). Thermal neu- 
trons; includes experimental work. 

B. T. Feld, Phys. Rev., 76, 1115 (1949). Fast neutrons. 

H. Feshbach, D. Peaslee, and V. Weisskopf, Phys. Rev., 71, 145, 564 
(1947). The basis for our version of the theory of dispersion. 



200 A Survey of Nuclear Reactions [Pt. VI 

Experimental 

R. K. Adair, Revs. Modern Phys., 22, 249 (1950). A comprehensive 

review. 
D. J. Hughes, W. Spatz, and N. Goldstein, Phys. Rev., 75, 1781 (1949). 

(d) Charged Particle Excitation 

Theoretical 

V. Weisskopf and D. Ewing, Phys. Rev., 57, 472, 935 (1940). 

D. Peaslee, Phys. Rev., 74, 1001 (1948). Deuterons only. 
Experimental 

E. L. Kelly, thesis, University of California, Berkeley, UCRL-1044, 1951. 
S. N. Ghoshal, Phys. Rev., 80, 939 (1950). 

P. C. Gugelot, Phys. Rev., 81, 51 (1951). 

(e) Nuclear Fission 

Theoretical 

N. Bohr and J. Wheeler, Phys. Rev., 56, 426 (1939). General theory, 

using drop model. 
D. Brunton, Phys. Rev., 76, 1798 (1949). 
Experimental 

Plutonium Project, Revs. Modern Phys., 18, 513 (1946). 
J. Jungerman and S. C. Wright, Phys. Rev., 76, 111 (1949). 
J. Jungerman, Phys. Rev., 79, 632 (1950). Alpha- and alpha-induced 
fission. 

(f) High-Energy Nuclear Stars 

Theoretical 

W. Horning and L. Baumhoff, Phys. Rev., 75, 370 (1949). 
M. L. Goldberger, Phys. Rev., 74, 1269 (1948). 
Y. Fujimoto and Y. Yamaguchi, Prog. Theor. Phys., 4, 468 (1950). 
Experimental 

G. Bernardini, E. Booth, and S. Lindenbaum, Phys. Rev., 80, 905 (1950). 

400-Mev protons. 
R. Menon, H. Muirhead, and O. Rochat, Phil. Mag., 41, 583 (1950). Pi- 
meson-induced' stars. 

Papebs on Specific Reactions 

The reactions are grouped by their means of excitation (list closed July, 1951). 
The references for each reaction are coded according to the first two or three letters 
in their authors' names and the years the works were published. A star (*) means 
that work above 100 Mev is reported. Some conventions are worth pointing out: 

1. Strictly speaking, gamma-rays often are secondary products of other reactions. 
These are not separately considered. Only reactions in which gammas are the sole 
product are listed under "Gamma-rays." 

2. Electron-induced reactions in which the electron is in fact captured by inverse 
beta-decay have not yet been observed. The electron serves to excite the nucleus, 
and itself passes on. It comes away from the reaction, though it is not in fact a 
reaction product. 

3. Reactions with more than two product particles are listed only once; the order 
of emission (if known) is not taken into account. 

The classification here adopted depends upon the facts: (i) for evident reasons of 
effectiveness and availability, the incident particles used in almost all nuclear reac- 



Pt. VI] Appendix II 201 

tions are light nuclei, up to helium (plus photons and electrons, etc.); and (ii) most 
reactions yield for at least one of the products another light nuclear particle, leaving 
behind as the second product particle a residual nucleus not far removed from the 
target nucleus in charge or mass number. 

Gamma-rays 

( T , T ) Sc46; Wi45b; Wi45c; Gu41; Gae49. 

( T , meson) *Lax51; *Lit51; *Mo50; *Pet51; *Ste50. 

( T , n) Bal46; *Per49; Mc49; By51; Jo50; Kat51b; Med50; Pri50. 

(7, 2«) Bal46; *Per48. 

( T , p) Same as (7, n) Hi47; Cou51; Di50; *Lev51; To51; *Wak51; By51. 

( T ; p, n, or d) Bal46; *Per48; By51. 

(7, a) Has51; Mil50; Pr50. 

(7, multiple products) Bal46; *Per48. 

(7, star) *Gae50. 

(7, fission) Ti49; Hax41; Bo39; Ko50; *Su50. 
Electrons 

(e; e', n) B149; Sk48. 

(e; e') Wi44; Wi45a; Wi45c; Mul51. 
Mesons 

O, it) *Ca51; *Be50. 

(*-, star) *Be50; *Che50; *Men50; *Ta50. 
Neutrons 

(n, 7) Fes47; *Kn49; Hug49; Gos47; An50; Cap51; Ham50; He50; Hu51; 
Mu50; Ki51. 

(n, n) Fe47; Fel49; Se49; Fes47; Ad49; Har50; *Pas50; Stt51. 

(n, In) Hou46; Ma42; Hey37; Coh51; Fow50; Waf50. 

(n, spallation) De48. 

(n, p) Coo49; Boo37; Am35; Sa40; Coh51. 

(n; p, n, or d) *Kn49; *Bru49; *Chw50; *Ha50; Waf50. 

(n; H 3 or He 3 ) Bru49; Cor41; On40; *Ha50. 

(n, a) Sh41; Wu40; *Kn49; Am35; Sa40; B046; St50. 

(n, stars) *Go48; *Tr50. 

(n, fission) P146; *Ke48; Ph49; Fra47; Bou50; Ros50. 
Protons 

(p, 7) Du38; Wal48; Ben46; Fo48; Dev49; Hal50. 

(p, t) *Bj50; *Blo51; *Hen51; *Jon50. 

(p, n) Wei40; Du38; Del39; Bla51; *Bod51; *Kn51; *Mir51. 

(p, In) Te47a. 

(p, spallation) *Baz50; *Baz51; *Hy50; *Me51. 

(p, p) Fu48; Ba39; Hei47; Bed49; Goh51; Lth50; Rh50. 

(p; p, n, or d) Ri46; Pa48; Th49. 

(p, a) Th49; Bur49; Bar50; Ch50; Coc49; Ra50; Dev49. 

(p, heavy particle) *Mar51; *Wr50. 

(p, star) *Tho49; *Cam50; *Fr50; *Hod51; *Pek50. 

(p, fission) Ju48; Bo39. 
Deuterons 

(d, 7) We43. 

(d, n) Pe48; Ke49; C146b; Ro47; Amm49; Fa49; *Kn51. 

(d, 2ra) Ke49; Te47a. 



202 



A Survey of Nuclear Reactions 



[Pt. VI 



(d, spallation) Wil48; *Mi48; *Bat50; *Lin50. 

(d, p) Pe48; Pol49; C146b; A1148; Buw50; Cu50; Hav51; Phi50; Va51. 

(d, pa) C147. 

(d, d) Gug47; Gr49; Ker51. 

(d, H 3 ) Kr41; Wi46; Ka49. 

(d, a ) Kri49; Li38; C146a; In50; Sch50. 

(d, an) Mad50. 

{d, stars) *Ga49a; *Ho49. 

(d, fission) *Wo49; Ju48; Kr40; *Goe49. 
Tritium 

(H 3 , n) Cr51. 

(H 3 , p) Ku48a. 

(H 3 , He 3 ) Ku47b. 
He 3 

(He 3 , p) A139. 
Alpha-particles 

(a, 7) Ben51. 

(a, mesons) *Bu49; *Jon51. 

(a, n) Eg48; Bra47; Rid37; Rie48; Hap49; Tem49. 

(a, 2n) Gh48; Te47a; Fi50; Tem49. 

(a, spallation) *Wo49; *Oc48; *Lin50. 

(a, p) Bro49; Roy51. 

(or, a) La39. 

(a; p, 3n) Hel46; Ne49; Te47b. 

(a, stars) *Ga49b. 

(a, fission) *Oc48; *Wo49; Ju48. 
Heavy particles 

(heavy particle, star) *Bra48; *Bra49. 



REFERENCES FOR APPENDIX II 

(Ad49) R. K. Adair, C. K. Bockelman, and R. E. Peterson, Phys. Rev., 76, 308 

(L) (1949). 
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(A1148) H. R. Allan and C. A. Wilkinson, Proc. Roy. Soc. (London), A194, 131 

(1948). 
(Am35) E. Amaldi, O. D'Agostino, E. Fermi, B. Pontecorvo, F. Rasetti, E. Segre, 

Proc. Roy. Soc. (London), A149, 522 (1935). 
(Amm49) P. Ammiraju, Phys. Rev., 76, 1421 (L) (1949). 
(An50) H. L. Anderson, Phys. Rev., 80, 499 (1950). 
(Ba39) S. Barnes and P. Aradine, Phys. Rev., 55 (1950). (In) 
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(Bar50) C. A. Barres, A. P. French, and S. Devons, Nature, 166, 145 (L) (1950). 
(Bat50) F. O. Bartell, A. C. Helmholz, S. D. Softky, and D. B. Stewart, Phys. 

Rev., 80, 1006 (1950). 
(Baz50) R. E. Batzel and G. T. Seaborg, Phys. Rev., 79, 528 (L) (1950). 
(Baz51) R. E. Batzel and G. T. Seaborg, Phys. Rev., 82, 607 (1951). 
(Be50) G. Bernardini, E. T. Booth, L. Lederman, and J. Tinot, Phys. Rev 

80, 924 (L) (1950). 



Pt. VI] 



Appendix II 



203 



(Bed49) R. S. Bender, F. C. Shoemaker, S. G. Kaufmann, and G. M. B. Bou- 

vicius, Phys. Rev., 76, 273 (1949). 
(Ben46) W. E. Bennett, T. W. Bonner, C. E. Mandeville, and B. E. Watt, Phys. 

Rev., 70, 882 (1946). 
(Ben51) W. E. Bennett, P. A. Roys, and B. J. Toppel, Phys. Rev., 82, 20 (1951). 
(Bj50) R. Bjorklund, W. E. Crandall, B. V. Moyer, and H. F. York, Phys. Rev., 

77, 213 (1950). 
(B149) J. S. Blair, Phys. Rev., 75, 907 (L) (1949). 

(Bla51) - V. P. Blaser, F. Boehm, P. Marmier, and D. C. Peaslee, Helv. Phys. 

Acta, 24, 3 (1951). 
(Blo51) M. M. Block, S. Passman, and W. W. Havens, Phys. Rev., 83, 167 (L) 

(1951). 
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(Bou50) C. C. Bounton and G. C. Hanna, Can. J. Research, 28, 498 (1950). 
(Bra47) H. L. Bradt and D. J. Tendam, Phys. Rev., 72, yi7 (1947). 
(Bra48) H. L. Bradt and B. Peters, Phys. Rev., 74, 1828 (1948). 
(Bra49) H. L. Bradt and B. Peters, Phys. Rev., 75, 1779 (L) (1949). 
(Bro49) J. E. Brolby, Jr., M. B. Sampson, and A. C. G. Mitchell, Phys. Rev., 

76, 624 (1949). 
(Bru49) K. Brueckner and W. M. Powell, Phys. Rev., 75, 1274 (L) (1949). 
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(Bur49) W. Burcham and W. Freeman, Phys. Rev., 75, 1756 (1949). 
(Buw50) H. Burrows, W. M. Gibson, and J. Rotblat, Phys. Rev., 80, 1095 (L) 

(1950). 
(By51) P. R. Byerly, Jr., and W. E. Stephens, Phys. Rev., 83, 54 (1951). 
(Ca51) M. Camac, D. R. Corson, R. M. Littauer, A. M. Shapiro, A. Silverman, 

R. R. Wilson, and W. M. Woodward, Phys. Rev., 82, 745 (1951). 
(Cam50) V. Camerini, P. H. Fowler, W. O. Lock, and M. Muirhead, Phil. Mag., 

41, 413 (1951). 
(Cap51) P. C. Capson and A. J. Verhoeve, Phys. Rev., 81, 336 (1951). 
(Ch50) C. Y. Chao, Phys. Rev., 80, 1035 (1950). 

(Che50) W. B. Cheston and L. J. B. Goldfarb, Phys. Rev., 78, 683 (1950). 
(Chw50) C. F. Chew and M. L. Goldberger, Phys. Rev., 77, 470 (1950). 
(C146a) E. T. Clarke and J. W. Irvine, Jr., Phys. Rev., 69, 680 (A) (1946). 
(C146b) E. T. Clarke, Phys. Rev., 70, 893 (1946). (Thick target yield survey) 
(C147) E. T. Clarke, Phys. Rev., 71, 187 (1947). (Al) 

(Coc49) W. Cochrane and A. H. Hester, Proc. Roy. Soc. (London), A199, 458 

(1949). 
(Coh51) B. L. Cohen, Phys. Rev., 81, 184 (1951). 

(Coo49) J. Coon and R. Nobles, Phys. Rev., 75, 1358 (1949). (He 3 , N 14 ) 
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(Cou51) E. P. Courant, Phys. Rev., 82, 703 (1951). 
(Cr51) R. W. Crews, Phys. Rev., 82, 100 (L) (1951). 
(Cu50) C. D. Curling and J. O. Newton, Nature, 166, 339 (1950). 



204 



A Survey of Nuclear Reactions 



[Pt. VI 



(De48) F. de Hoffmann, B. T. Feld, and P. R. Stein, Phys. Rev., 74, 1330 (1948). 

(U 235 ) 
(Del39) L. A. Delsasso, L. N. Ridenour, R. Sherr, and M. G. White, Phys. Rev., 

55, 113 (1939). 
(Dev49) S. Devons and M. G. N. Hine, Proc. Roy. Soc. (London), A199, 56, 73 

(1949). 
(Di50) B. C. Diven and G. M. Almy, Phys. Rev., 80, 407 (1950). 

(Du38) L. DuBridge, S. Barnes, J. Buck, and C. Strain, Phys. Rev., 53, 447 

(1938). 
(Eg48) D. T. Eggen and M. L. Pool, Phys. Rev., 74, 57 (1948). 
(Fa49) D. E. Falk, E. Creutz, and F. Seitz, Phys. Rev., 76, 322 (L) (1949). 
(Fe47) E. Fermi and L. Marshall, Phys. Rev., 71, 666 (1947). (Theory and 

experimental survey) 
(Fel49) B. T. Feld, Phys. Rev., 75, 1115 (1949). (Theory and survey) 
(Fes47) H. Feshbach, D. C. Peaslee, and V. F. Weisskopf, Phys. Rev., 71, 145, 

564 (1947). 
(Fi50) R. W. Fink, F. L. Reynolds, and D. H. Templeton, Phys. Rev., 77, 614 

(1950). 
(Fo48) W. A. Fowler, C. C. Lauritsen, and T. Lauritsen, Revs. Modern Phys., 

20, 236 (1948). 
(Fow50) J. L. Fowler and J. M. Slye, Jr., Phys. Rev., 77, 787 (1950). 
(Fr50) P. Freier and E. P. Ney, Phys. Rev., 77, 337 (1950). 

(Fra47) S. Frankel and N. Metropolis, Phys. Rev., 72, 914 (1947). 
(Fu48) H. Fulbright and R. Bush, Phys. Rev., 74, 1323 (1948). (Light nuclei) 
(Ga49a) E. Gardner and V. Peterson, Phys. Rev., 75, 364 (1949). (d, stars in 

photoplates exp.) 
(Ga49b) E. Gardner, Phys. Rev., 75, 379 (1949). (Stars in photoplates exp.) 
(Gae49) E. R. Gaerttner and M. L. Yeater, Phys. Rev., 76, 363 (1949). 
(Gae50) E. R. Gaerttner and M. L. Yeater, Phys. Rev., 77, 714 (L) (1950). 
(Gh48) S. N. Ghoshal, Phys. Rev., 73, 417 (L) (1948). 
(Go48) M. L. Goldberger, Phys. Rev., 74, 1269 (1948). (Heavy nuclei: 

theory) 
(Goe49) R. H. Goeckermann and I. Perlman, Phys. Rev., 76, 628 (1949). 
(Goh51) G. Goldhaber and R. M. Williamson, Phys. Rev., 82, 495 (1951). 
(Gos47) H. H. Goldsmith, H. W. Ibser, and B. T. Feld, Revs. Modern Phys., 19, 

259 (1947). (Excitation survey) 
(Gr49) G. W. Greenlees, A. E. Kempton, and E. H. Rhoderick, Nature, 164, 663, 

(L) (1949). 
(Gu41) E. Guth, Phys. Rev., 59, 325 (1941). (Theory) 
(Gug47) K. M. Guggenheim, H. Heitler, and G. F. Powell, Proc. Roy. Soc. (Lon- 
don), A190, 196 (1947). 
(Ha50) J. Hadley and H. York, Phys. Rev., 80, 345 (1950). 
(Hal50) R. H. Hall and W. A. Fowler, Phys. Rev., 77, 197 (1950). 
(Hap49) I. Halpern, Phys. Rev., 76, 248 (1949). 
(Ham50) B. Hamermesh, Phys. Rev., 80, 415 (1950). 
(Har50) S. P. Harris, C. O. Muehlhause, and G. E. Thomas, Phys. Rev., 79, 11 

(1950). 
(Has51) R. N. H. Haslam and H. M. Skarsgard, Phys. Rev., 81, 479 (L) (1951). 
(Hav51) J. A. Harvey, Phys. Rev., 81, 353 (1951). 
(Hax41) R. O. Haxby, W. E. Shoupp, W. E. Stephens, and W. H. Wells, Phys. 

Rev., 59, 57 (1941). 



Pt. VI] 



Appendix II 



205 



(He50) R. L. Henkel and H. H. Barschall, Phys. Rev., 80, 145 (1950). 

(Hei47) H. Heitler, A. N. May, and C. P. Powell, Proc. Roy. Soc. (London), 

A190, 180 (1947). 
(Hel46) A. C. Helmholz, Phys. Rev., 70, 982 (L) (1946). 
(Hen51) E. M. Henley and R. H. Huddlestone, Phys. Rev., 82, 754 (1951). 
(Hey37) F. A. Heyn, Physiea, 4, 1224 (1937). 

(Hi47) O. Hirzel and H. Waffler, Helv. Phys. Acta, 20, 373 (1947). 

(Ho49) W. Horning and L. Baumhoff, Phys. Rev., 75, 470 (1949). (Theory, 

d, stars in photoplates) 
(Hod51) P. E. Hodgson, Phil. Mag., 42, 82 (1951). 
(Hou46) F. G. Houtermans, Nachr. Akad. Wiss. Gottingen, Math.-physik. Kl., 

1, 52 (1946). 
(Hu51) H. Hurwitz, Jr., and H. A. Bethe, Phys. Rev., 81, 898 (L) (1951). 
(Hug49) D. J. Hughes, W. D. B. Spatz, and N. Goldstein, Phys. Rev., 76, 1781 

(1949). 
(Hy50) E. K. Hyde, A. Ghiorso, and G. T. Seaborg, Phys. Rev., 77, 765 (1950). 
(In50) D. R. Inglis, Phys. Rev., 78, 104 (1950). 

(Jo50) H. E. Johns, L. Katz, R. A. Douglas, and R. N. H. Harlan, Phys. Rev., 

80, 1062 (1950). 
(Jon50) S. B. Jones and R. S. White, Phys. Rev., 78, 12 (1950). 
(Jon51) S. B. Jones and R. S. White, Phys. Rev., 82, 374 (1951). 
(Ju48) J. Jungerman and S. C. Wright, Phys. Rev., 74, 150 (1948). 

(Ka49) D. Kahn and G. Groetzinger, Phys. Rev., 75, 906 (L) (1949). 
(Kat51b) L. Katz, H. E. Johns, R. G. Baker, R. N. H. Harlan, and R. A. Douglas, 

Phys. Rev., 82, 271 (L) (1951). 
(Ke48) E. L. Kelly and C. Wiegand, Phys. Rev., 73, 1135 (1948). 
(Ke49) E. L. Kelly and E. Segre, Phys. Rev., 75, 999 (1949). (Bi) 
(Ker51) K. K. Keller, J. B. Niedner, C. F. Way, and F. B. Shull, Phys. Rev., 81, 

481 (L) (1951). 
(Ki51) B. B. Kinsey, G. A. Bartholomew, and W. A. Walker, Phys. Rev., 82, 

380 (1951). 
(Kn49) W. J. Knox, Phys. Rev., 75, 537 (1949). (Light nuclei) 
(Kn51) W. J. Knox, Phys. Rev., 81, 687 (1951). 
(Ko50) H. W. Koch, J. McElhinney, and E. L. Gasteiger, Phys. Rev., 77, 329 

(1951). 
(Kr40) R. S. Krishnan and T. E. Banks, Nature, 145, 860 (1940). 
(Kr41) R. S. Krishnan, Nature, 148, 407 (L) (1941). 

(Kri49) N. Krisberg and M. L. Pool, Phys. Rev., 75, 1693 (1949). (Ti) 
(Ku47b) D. Kundu and M. L. Pool, Phys. Rev., 72, 101 (1947). (Ag) 
(Ku48a) D. Kundu and M. L. Pool, Phys. Rev., 73, 22 (1948). (Rh and Co) 
(La39) K. Lark-Horowitz, J. R. Risser, and R. N. Smith, Phys. Rev., 55, 878 

(1939). (In) 
(Lax51) M. Lax and H. Feshbach, Phys. Rev., 81, 189 (1951). 
(Lev51) C. Levinthal and A. Silverman, Phys. Rev., 82, 822 (1951). 
(Li38) J. Livingood and G. Seaborg, Phys. Rev., 54, 391 (1938). 

(Lin50) M. Lindner and I. Perlman, Phys. Rev., 78, 499 (1950). 
(Lit51) R. M. Littauer and D. Walker, Phys. Rev., 82, 746 (1951). 
(Lth50) C. Levinthal, E. A. Martinelli, and A. Silverman, Phys. Rev., 78, 199 

(1950). 
(Ma42) W. Maurer and W. Raums, Z. Physik, 119, 602 (1942). 



206 



A Survey of Nuclear Reactions 



[Pt. VI 



(Mad49) C. E. Mandeville, C. P. Swann, and S. C. Snowdon, Phys. Rev., 76, 980 

(D (1949). 
(Mar51) L. Marquez and I. Perlman, Phys. Rev., 81, 953 (1951). 
(Mc49) J. McElhinney et al, Phys. Rev., 76, 542 (1949). (Gamma-excitation 
survey) 

(Mcd50) B. D. McDaniel, R. L. Walker, and M. B. Stearns, Phys. Rev., 80, 807 
(1950). 

(Me51) J. W. Meadows and R. B. Holt, Phys. Rev., 83, 47 (1951). 

(Men50) M. G. K. Menon, H. Muirhead, and O. Roehat, Phil. Mag., 41 (4th 
series), 583 (1950). 

(Mi48) D. Miller, R. Thompson, and B. Cunningham, Phys. Rev., 74, 347 (L) 
(1948). 

(Mil50) C. H. Millar and A. G. W. Cameron, Phys. Rev., 78, 78 (L) (1950). 

(Mir51) R. D. Miller, D. C. Sewell, and R. W. Wright, Phys. Rev., 81, 374 (1951). 

(Mo50) R. F. Mozley, Phys. Rev., 80, 493 (L) (1950). 

(Mu50) C. O. Muehlhause, Phys. Rev., 79, 277 (1950). 

(Mul51) C. J. Mullin and E. Guth, Phys. Rev., 82, 141 (1951). 

(Ne49) A. Newton, Phys. Rev., 75, 209 (L) (1949). 

(Oc48) P. R. O'Connor and G. Seaborg, Phys. Rev., 74, 1189 (L) (1948). 

(On40) R. D. O'Neal and M. Goldhaber, Phys. Rev., 58, 574 (1940). (Li) 

(Pa48) W. K. Panofsky and R. Phillips, Phys. Rev., 74, 1732 (L) (1948). 

(Pas50) S. Pasternack and H. S. Snyder, Phys. Rev., 80, 921 (L) (1950). 

(Pe48) D. C. Peaslee, Phys. Rev., 74, 1001 (1948). (Theory, d-excitation) 

(Pek50) D. H. Perkins, Proc. Roy. Soc. (London), 203, 399 (1950). 

(Per48) M. Perlman and G. Friedlander, Phys. Rev., 74, 442 (1948). (Gamma- 
excitation survey) 

(Per49) M. Perlman, Phys. Rev., 76, 988 (1949). (Table) 

(Pet51) J. M. Peterson, W. S. Gilbert, and R. S. White, Phys. Rev., 81, 1003 
(1951). 

(Ph49) A. Phillips, L. Rosen, and R. Taschek, Phys. Rev., 75, 919 (1949). 

(Phi50) G. C. Phillips, Phys. Rev., 80, 1(34 (1950). 

(PI46) Plutonium Project, Revs. Modern Phys., 18, 513 (1946). (Fission 

survey) 

(Pol49) E. Pollard, V. Sailor, and L. Wyly, Phys. Rev., 75, 725 (1949). (Al) 

(Pr50) M. A. Preston, Phys. Rev., 80, 307 (L) (1950). 

(Pri50) G. A. Price and D. W. Kerst, Phys. Rev., 77, 806 (1950). 

(Ra50) J. K. Rasmussen, W. F. Hornyak, C. C. Lauritsen, and T. Lauritsen, 
Phys. Rev., 77, 617 (1950). 

(Rh50) E. H. Rhoderick, Proc. Roy. Soc. (London), A201, 348 (1950). 

(Ri46) J. R. Richardson and B. T. Wright, Phys. Rev., 70, 445 (A) (1946). 

(Rid37) L. N. Ridenour and W. J. Henderson, Phys. Rev., 52, 889 (1937). (Ex- 
citation survey) 

(Ric48) H. T. Richards, MDDC 1504. 

(Ro47) R. B. Roberts and P. H. Abelson, Phys. Rev., 72, 76 (1947). 

(Ros50) L. Rosen and A. M. Hudson, Phys. Rev., 78, 533 (1950). 

(Roy51) R. R. Roy, Phys. Rev., 82, 227 (1951). 

(Sa40) R. Sagane, S. Kojima, G. Miyamoto, M. Ikawa, Phys. Rev., 57, 1179 

(L) (1940). 

(Sc46) L. I. Schiff, Phys. Rev., 70, 761 (1946). 

(Sch50) A. D. Schelberg, M. B. Sampson, and R. G. Cochran, Phys. Rev. 80, 
574 (1950). 



Pt. VI] 



Appendix II 



207 



(Se49) F. G. P. Seidl, Phys. Rev., 75, 1508 (1949). 

(Sh41) R. Sherr,,K. Bainbridge, and H. Anderson, Phys. Rev., 60, 473 (1941). 

(Pt, Hg) 
(Sk48) L. S. Skaggs, J. S. Laughlin, A. O. Hanson, and J. J. Orlin, Phys. Rev., 

73, 420 (L) (1948). 
(St50) A. Stebler, H. Bichsel, and P. Huber, Helv. Phys. Acta, 23, 511 (1950). 

(Ste50) J. Steinberger, W. K. H. Panofsky, and J. Steller, Phys. Rev., 78, 802 

(1950). 
(Stt51) P. H. Stetson and C. Goodman, Phys. Rev., 82, 69 (1951). 
(Su50) N. Sugarman, Phys. Rev., 79, 532 (L) (1950). 

(Sz48) A. Szalay and E. Csongor, Phys. Rev., 74, 1063 (1948). (Mg) 

(Ta50) S. Tamor, Phys. Rev., 77, 412 (L) (1950). 
(Te47a) D. H. Templeton, J. J. Howland, and I. Perlman, Phys. Rev., 72, 758 

(1947). 
(Te47b) D. H. Templeton, J. J. Howland, and I. Perlman, Phys. Rev., 72, 766 

(1947). 
(Tem49) G. M. Temmer, Phys. Rev., 76, 424 (1949). 
(Th49) R. Thomas, S. Rubin, W. Fowler, and C. Lauritsen, Phys. Rev., 76, 1612 

(L) (1949). 
(Tho49) G. Thomson, Phil. Mag., 40, 589 (1949). 

(Ti49) E. W. Titterton and F. K. Goward, Phys. Rev., 76, 142 (L) (1949). 

(Ti51) E. W. Titterton, Phil. Mag., 42, 109 (L) (1951). 

(To51) M. E. Toms and W. E. Stephens, Phys. Rev., 82, 709 (1951). 

(Tr50) J. Tracy and W. M. Powell, Phys. Rev., 77, 594 (1950). 

(Va51) E. M. Van Patler, W. W. Buechner, and H. Sperduto, Phys. Rev., 82, 

248 (1951). 
(Waf50) H. Waffler, Helv. Phys. Acta, 23, 239 (1950). 
(Wak51) D. Walker, Phys. Rev., 81, 634 (L) (1951). 
(Wal48) R. L. Walker and B. D. McDaniel, Phys. Rev., 74, 315 (1948). 
(We43) K. Weiner, M. L. Pool, and J. Kurbatov, Phys. Rev., 63, 67 (1943). 
(Wei40) V. Weisskopf and D. Ewing, Phys. Rev., 57, 472, 935 (1940). (Theory 

and experiment: p, n.) 
(Wi44) M. L. Wiedenbeck, Phys. Rev. 
(Wi45a) M. L. Wiedenbeck, Phys. Rev., 
(Wi45b) M. L. Wiedenbeck, Phys. Rev. ; 
(Wi45c) M. L. Wiedenbeck, Phys. Rev. 
(Wi46) M. L. Wiedenbeck, Phys. Rev., 70, 435 (L) (1946). 
(Wil48) G. Wilkinson and H. Hicks, Phys. Rev., 74, 1733 (L) (1948). (Table, 

rare earths) 
(Wo49) R. D. Wolfe and N. E. Ballou, Phys. Rev., 75, 527 (L) (1949). 
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(Wr50) S. C. Wright, Phys. Rev., 79, 838 (1950). 



, 66, 36 (A) (1944). 
, 67, 59 (A) (1945). 
, 68, 1 (1945). 
, 68, 237 (1945). 



PART VII 



The Neutron 

BERNARD T. FELD 

The Massachusetts Institute of Technology 

In 1948, when this work was begun, the status of the field of neutron 
physics was uncertain and rather anomalous. Important progress, 
which had been made during the war, was known to the great body of 
physicists only through a few "releases" and through the Smyth report. 
The main prewar references — the articles of Bethe, Bacher, and Liv- 
ingston, in Reviews of Modern Physics — were hopelessly out-of-date. 
They had been partially and inadequately replaced by hastily assembled 
and informally distributed mimeographed notes of a Los Alamos lecture 
series on nuclear physics (LA-24) and of a series of lectures by Fermi on 
neutron physics. 

By the time the first draft of this work was completed, in June of 
1949, the situation was quite a different one. As a consequence of a 
wise and far-sighted policy on the part of the American, British, and 
Canadian atomic energy commissions, practically all the basic scientific 
data, which had been accumulated during the war, appeared in the open 
literature. As a result of the widespread renewal of interest in the prob- 
lems of neutron physics, the field has developed, and continues to de- 
velop, at a rate which has converted the task of compiler and author 
into an almost hopeless struggle against obsolescence and has expanded 
the dimensions of this work far beyond its original conception. The 
unequal struggle was, quite arbitrarily, concluded as of July 1, 1951. 
Since then, some changes have been made, mainly as a result of impor- 
tant new developments which have come to the attention of the author, 
mostly through publication in American journals. 

This work is primarily intended for the practicing nuclear physicist. 
It presupposes a knowledge of the fundamentals of nuclear physics as 
well as of its terminology. Some of the terms, such as "barn," 1 are of 
comparatively recent origin, but their use has become widespread. 

1 Barn, a unit of cross section; origin: big as a . (1 barn = 10 -24 cm 2 .) 

This unit is said to have been invented by the nuclear physicists at Purdue Uni- 
versity, around 1941 or 1942, to describe nuclear cross sections which are relatively 
easy to measure. 

208 



Sec. 1A] Properties and Fundamental Interactions 209 

Although an attempt has been made to present herein a complete 
summary of the major aspects of neutron physics, some important 
applications have, perforce, received inadequate treatment or been 
omitted completely. Thus, no attention has been paid to the role of 
neutrons in biophysics; nor has any space been devoted to the many 
interesting problems concerning the effects of neutron interactions on 
the macroscopic physical properties of matter. Also, and regretfully, 
omitted is a discussion of the fascinating problems of the origin of the 
elements, in which the properties of neutron interactions may have 
played a decisive role. [The interested reader is referred to an excellent 
summary of this field by Alpher and Herman, Revs. Mod. Phys., 22, 153 
(1950).] 

It would be impossible to give a complete list of the many individuals 
to whom I am indebted for aid, encouragement, information, discussion, 
and criticism during the progress of this work. My colleagues at M.I.T., 
Brookhaven, and elsewhere, especially H. Feshbach, M. Goldhaber, 
G. Placzek, and V. F. Weisskopf , have been unusually generous in this 
respect. I owe a special debt to E. Amaldi, of the University of Rome, 
not only for congenial and informative discussions but for having gen- 
erously allowed me access to an unpublished work on neutron physics 
by G. C. Wick and himself. 

In memory of many pleasant hours, during which the outline of this 
article achieved shape and substance, I respectfully dedicate this work 
to my teacher and friend, the late H. H. Goldsmith. 

SECTION 1. PROPERTIES AND FUNDAMENTAL 
INTERACTIONS 

A. Discovery 

The discovery of the neutron is one of the most dramatic chapters in 
the history of modern physics. It started in Germany, in the last month 
of 1930, with the report by Bothe and Becker (B50) of a penetrating 
radiation resulting from the bombardment of certain light elements by 
polonium alpha-particles. They bombarded many substances, using a 
Geiger point-counter as a detector of the resulting radiation. Most of 
the elements investigated (Pb, Ag, Ca, N, C, O, Ne) yielded no detect- 
able radiation; two (Mg, Al) showed a slight effect. Lithium, boron, 
and fluorine gave appreciable amounts of radiation capable of affecting 
the counter, and beryllium yielded a comparatively tremendous amount. 
Bothe and Becker concluded that the radiation consisted of gamma- 



210 The Neutron [Pt. VII 

rays, more penetrating than any that had been observed up to that 
time. 1 

Curie and Joliot (C36, J10) immediately undertook a study of the 
properties of this penetrating radiation. They had available a much 
stronger polonium source (100 millicuries, as compared to the 3-7 
available to Bothe and Becker) and were able to measure the absorp- 
tion of the radiation in lead. They observed, for the radiation from 
beryllium, an exponential attenuation with an absorption coefficient of 
0.15 cm -1 . (The radiations from boron and lithium had lead absorp- 
tion coefficients of 0.2 and 1.7 cm -1 , respectively.) 

So far, there was nothing to contradict the suggestion — and, indeed, 
this was universally assumed — that the radiations were very penetrat- 
ing gamma-rays. (We know now that an absorption coefficient in lead 
of 0.15 cm -1 is smaller than that of the most penetrating gamma-rays.) 
However, in January of 1932, Curie and Joliot (C37) reported the fol- 
lowing interesting observations : They investigated the effects of placing 
thin screens of various materials in front of the ionization chamber, 
which was being used to detect the radiations. For most of the screens 
nothing noteworthy occurred. However, when the screens contained 
hydrogen, the current in the ionization chamber went up. 

Curie and Joliot inferred that the increased ionization was due to the 
ejection of protons from the screen by the primary radiation. They 
strengthened this theory by the following set of observations: (1) The 
application of a magnetic field in the region between the screen and the 
ionization chamber did not decrease the effect; it would have, if the 
ejected particles had been slow electrons. (2) The effect vanished when 
0.2 mm of aluminum was placed between the screen and the chamber. 
This was sufficient to absorb protons, but not fast electrons. (3) Cloud 
chamber photographs of the tracks of particles ejected by the radiation 

1 The conclusions of Bothe and Becker, as well as the observations upon which 
they were based, turn out, in retrospect, to be completely valid. The Geiger point- 
counter, used as a detector by these investigators, was not sensitive to neutrons; 
thus, Bothe and Becker could not have observed neutrons in their experiments. It 
was only with the introduction of ionization chambers, proportional counters, and 
cloud chambers as detectors that the neutrons became observable in subsequent 
investigations. The measurements of the gamma-ray energies (~3 Mev from boron 
and ~5 Mev from beryllium), reported by Bothe and Becker, have subsequently 
been confirmed. Indeed, the important discovery by Bothe and Becker of artificial 
excitation of nuclear gamma-radiation, reported in their 1930 paper, has unfortu- 
nately been almost completely obscured by the drama associated with the discovery 
of the neutron. For a more complete discussion of the significance of the observa- 
tions of Bothe and co-workers, the reader is referred to an article by Fleischmann 
(F32). 



Sec. 1A] Properties and Fundamental Interactions 211 

from hydrogen-containing screens showed ionization consistent with 
that of protons, but inconsistent with that of electrons (C38). 

Curie and Joliot first hypothesized that the ejection of protons from 
the screens was due to Compton scattering of the incident gamma-radia- 
tion by the hydrogen nuclei. From the observed range (energy) of the 
(recoil) protons, they estimated the energy of the photons from beryl- 
lium to be 50 Mev. Curie and Joliot recognized a number of serious 
difficulties in connection with their hypothesis; they could conceive of 
no source of such high-energy photons in a reaction of alpha-particles on 
beryllium, and they felt (erroneously) that photons of such high energy 
should be even more penetrating than the observed radiation. But, 
while they decided that the ejection of protons (and other light nuclei) 
was by a new type of gamma-ray interaction (C38), it remained for 
Chadwick, working at the Cavendish Laboratory in England, to reject 
the gamma-ray hypothesis and take the bold step of postulating a new 
particle. 

Chadwick was very quick to follow up the researches reported from 
the Institut du Radium. Only slightly more than a month after the 
report discussed above he was publishing data (C5), obtained with 
counters and cloud chambers, showing that the radiations from beryl- 
lium bombarded with alpha-particles were capable of conferring high 
speeds not only upon protons, but also upon the nuclei of other light 
elements (He, Li, Be, B, C, N, 0, A). From the observed ranges of the 
light nuclei, and using the then current range-energy relationships, 
Chadwick showed that the Compton recoil hypothesis of Curie and 
Joliot was inconsistent with the data. The data could, however, be 
explained if the light nuclei were assumed to be recoils from elastic 
collisions with a neutral (to explain the great penetrability) particle 
of approximately protonic mass. 

To obtain the mass of this particle (called by Chadwick the neutron) 
Chadwick (C6) used the available data on the maximum range (velocity) 
of the proton recoils, and the results of Feather (F2) on the maximum 
range of the nitrogen recoils, observed in a cloud chamber. (If these 
were due to Compton recoils, they would have required gamma-ray 
energies of 55 and 90 Mev, respectively.) By application of the law of 
conservation of momentum, Chadwick derived that the particles re- 
sponsible for these recoils had a mass of 1.15 times the proton mass, with 
an uncertainty such that "it is legitimate to conclude that the mass of 
the neutron is very nearly the same as the mass of the proton." 

Another estimate of the mass of the neutron was made by Chadwick 
from observations on the neutron-producing reaction B 11 + He 4 — > N 14 
+ n 1 . On the assumption that the maximum-energy neutrons (obtained 



212 The Neutron [Pt. VII 

from the maximum range of proton recoils) correspond to leaving N 14 in 
its ground state, and from the values of the masses of the three nuclei 
involved, previously determined by Aston, Chadwick deduced the value 
of 1.0067 atomic mass units for the mass of the neutron. 

Within a short time, immediately following upon the series of an- 
nouncements described above, a large number of investigators in many 
lands were conducting experiments on the properties of the neutron and 
its interactions. Although it is not the author's intention to continue 
this historical survey much beyond the discovery stage, it is of some 
interest to review briefly the advances made within only a year after 
the appearance of Chadwick's papers. For a more detailed discussion 
of the early history, the reader should refer to the original papers and 
to excellent review articles (written in 1933) by Chadwick (C7) and 
Darrow (D4). 

It was soon ascertained that the neutrons resulting from the bom- 
bardment of beryllium and boron by polonium alpha-particles had a 
rather wide energy spread (F2, M21, C39) including many neutrons of 
energy considerably below 1 Mev (A36), with the neutron energy 
strongly dependent on the direction of emission, with respect to the 
alpha-particle direction, as well as on the energy of the alpha-particle. 
Gamma-rays were also shown to be emitted in the neutron-producing 
reaction (B51, C39). 

The yield of neutrons from beryllium and boron was found to decrease 
rapidly with decreasing alpha-particle energy (R7, C40, C7). 

By placing considerable quantities of lead next to the neutron-detect- 
ing ionization chamber (but not directly in the beam) (B70) or by sur- 
rounding the cloud chamber with copper (A36), the number of neutrons 
detected was appreciably increased, thus indicating a large neutron 
scattering by these substances. 

In the nitrogen gas of the cloud chamber, used in the experiments of 
Feather (F2), a number of events were observed which could be as- 
cribed to the transmutation N 14 + n 1 — ► B 11 + He 4 , the reverse of 
the reaction on boron which had been observed to produce neutrons. 
He also observed transmutations in oxygen and carbon (F3). Transmu- 
tations were observed in nitrogen, oxygen, and aluminum by Meitner 
and Philipp (M21), and in nitrogen by Harkins, Gans, and Newson 
(H37) and by Kurie (K27). 

At first, it was generally assumed that the neutron is probably a 
closely bound combination of a proton and an electron, 1 especially since 

1 As early as 1920, Rutherford (R27) published an interesting and prophetic specu- 
lation: "Under some conditions ... it may be possible for an electron to combine 
much more closely with the H nucleus [than in the neutral hydrogen atom], forming 



Sec. IB] Properties and Fundamental Interactions 213 

the first estimates indicated that its mass is less than that of the proton. 
The first suggestion that the neutron should be regarded as a fundamen- 
tal particle appears to have come from Iwanenko (13), and it soon 
became clear, mainly on the basis of quantum-mechanical arguments 
involving the spin and statistics of light nuclei, that the neutron, like 
the proton, probably has a spin of J and obeys Fermi-Dirac statistics 
(M21, C7). Thus, with the discovery of the neutron and the recogni- 
tion of its properties, the currently accepted picture of nuclei, as con- 
sisting of protons and neutrons, soon emerged. 

Since 1933 the development of the field has proceeded with rapidly 
increasing intensity. We therefore abandon, at this point, the historical 
survey. Instead, we shall summarize and discuss the present knowledge 
of the properties of neutrons and their interactions. We shall, however, 
in discussing each aspect, attempt to include some of the historical back- 
ground. It must be emphasized that, although, in the relatively short 
time that has elapsed since the discovery of the neutron, tremendous 
progress has been made in understanding and utilizing it, there is still 
much to be done before the neutron can be said to be completely under- 
stood. Many important and crucial experiments are in the process of 
being performed, while others are still in the future. 

B. Properties 

Since the neutron is one of the constituents of atomic nuclei, a knowl- 
edge of its properties . is fundamental for the understanding of nuclei. 
In addition, the properties of the neutron determine, to a large extent, 
the interactions between neutrons and nuclei, and between neutrons and 
conglomerations of nuclei (matter). In this section we summarize 
these properties from the point of view of the neutron as a fundamental 
particle. 

Wherever possible, we shall discuss the experimental evidence on 
which the conclusions as to the nature of the neutron are based. How- 
ever, the understanding of many of these experiments depends on a 
detailed knowledge of the interaction of neutrons with matter. In such 
cases, the results of the experiments will simply be stated, and the dis- 
cussion of the experiments reserved for subsequent sections. 

a kind of neutral doublet. Such an atom would have very novel properties. Its 
external field would be practically zero, except very close to the nucleus, and in 
consequence it should be able to move freely through matter. Its presence would 
probably be difficult to detect by the spectroscope, and it may be impossible to 
contain it in a sealed vessel. On the other hand, it should enter readily the struc- 
ture of atoms, and may either unite with the nucleus or be disintegrated by its 
intense field, resulting possibly in the escape of a charged H atom or an electron 
or both." 



214 The Neutron [Pt. VII 

Before proceeding with the discussion of the neutron, a few points 
should be noted in justification of devoting so much attention to the 
neutron as compared to other nuclear particles. The neutron is un- 
matched among atomic and nuclear particles as a tool for the investiga- 
tion of nuclear properties. The distinguishing feature of neutrons is 
the absence of electric charge; thus, the interaction of neutrons with 
matter is primarily determined by purely nuclear properties. 

In the fifty or so years since its beginnings, nuclear physics has made 
large contributions to the development of the other sciences (chemistry, 
geology, biology, medicine) and to technology. Since its discovery the 
neutron has assumed a most important role in furthering this progress. 
Were there no other reason, the importance of the neutron for the nu- 
clear chain reaction might constitute sufficient justification for assign- 
ing to it a position of special significance among the tools of modern 
science. 

1. Charge. The neutron is usually assumed to have no net charge. 
This assumption is consistent with all the observed properties and inter- 
actions of neutrons. However, the observations do not preclude the 
possibility that the neutron may have a net charge so small as to have 
heretofore eluded detection. It is therefore of interest to derive from 
the available evidence an upper limit for the magnitude of the possible 
neutron charge. 

The most direct evidence on the neutron's neutrality comes from the 
experiments of Dee (D6), reported at the same time as Chadwick's 
announcement of the discovery of the neutron. He investigated the 
ionization produced in air in a cloud chamber irradiated by fast neu- 
trons and concluded that, if the neutron interacts with atomic electrons 
at all, this process produces not more than one ion pair per 3 meters of 
the neutron's path in air. From these data it may be concluded (F36) 
that the charge of the neutron is less than J^oo of the proton charge. 

A somewhat less direct determination of an upper limit to the neutron 
charge may be obtained from considerations involving the observed neu- 
trality of atoms throughout the periodic table. Since the ratio of neu- 
trons to protons in atomic nuclei increases from zero in hydrogen to 1.6 
in uranium, this neutrality implies both that the difference between the 
proton and electron charges is small and that the neutron charge is 
small. An estimate of the, possible magnitude of the neutron charge 
depends on the accuracy with which we know atoms to be neutral. 
Rabi J and co-workers have observed that the molecule Csl (108 pro- 
tons and electrons and 152 neutrons) has a net charge less than 10 — 10 of 

1 Private communication. 



Sec. IB] Properties and Fundamental Interactions 215 

the electron charge. From this observation it is possible to draw a num- 
ber of conclusions: (1) The smallness of the charge of Csl may be due to 
an accidental cancellation of the neutron charge and the proton-electron 
charge difference. In this case, which is exceedingly unlikely, the mag- 
nitude of these charges can only be determined by another observation 
on some other atom or molecule, with a different neutron-proton ratio. 
In any event, both the neutron charge and the proton-electron charge 
difference would have to be quite small to account for the neutrality of 
atoms, say < 10 — 5 e. (2) Either the neutron charge or the proton-elec- 
tron charge difference is zero (or considerably less than 10~~ 12 electron 
charges), in which case the non-zero charge is less than 10 — 12 electron 
charges. (3) The neutron charge and the proton-electron charge dif- 
ference are both finite, and very small. Thus, if they were equal and 
opposite, the neutron charge would have to be less than 2 X 10 — 12 
electron charges. It seems quite reasonable to conclude, from this evi- 
dence, that the net charge on the neutron is exceedingly small, probably 
less than 10 — 12 electron charges. 

Despite the electrically neutral character of the neutron, there is a 
very small electromagnetic interaction between neutrons and charged 
particles arising out of the magnetic dipole moment of the neutron (to 
be discussed). In addition, it should be pointed out that, according to 
currently prevailing theories, the neutron should not strictly be regarded 
as a fundamental particle, but rather as having a complex structure 
involving equal numbers of + and — charged particles (mesons). 
Such a structure is certainly required for the understanding of the mag- 
netic moment of the neutron. The complex nature of the neutron would 
lead to a small, purely electrical interaction between neutrons and elec- 
trons; the existence and magnitude of such a neutron-electron interac- 
tion is being investigated and is discussed further on. 

Two other points are worth noting as having a bearing on the possible 
charge of the neutron: (1) The observation of the decay of the neutron 
into a proton, an electron, and a neutrino (discussed in Section 1B3) 
implies that (a), if the proton and electron charges are equal and oppo- 
site, the neutron and the neutrino have the same charge, if any; or (b), 
if the neutrino is uncharged, the neutron charge is equal to the proton- 
electron charge difference; or (c) some combination of (a) and (b). In 
this connection, de Broglie has pointed out (B69) that hypothesis (a) 
would be consistent with the neutrality of atoms if there were both posi- 
tively and negatively charged neutrinos and, correspondingly, neutrons, 
so that heavy nuclei could contain approximately equal numbers of neu- 
trons of the two charges. (2) The smallness of the neutron-electron 



216 The Neutron [Pt. VII 

interaction (Section 1C3) implies that the charge on the neutron must 
be exceedingly small. In particular, the observation by Fermi and Mar- 
shall (F23) that the scattering of thermal neutrons by xenon atoms is 
spherically symmetrical to better than 1 percent allows us to set an 
upper limit on the possible net charge of the neutron of >~10~ 18 electron 
charges. 

2. Mass. 1 The most accurate determination of the neutron mass is 
obtained, indirectly, from observations on nuclear transmutations in 
which all the masses and energies, except the mass of the neutron, are 
known. Thus, the estimate of Chadwick, quoted above, was obtained 
from consideration of the B u (a,n)N u reaction (C7). In the same 
paper Chadwick observed that a more direct determination of the neu- 
tron mass could be obtained if the binding energy of the deuteron, the 
nucleus consisting of a neutron and a proton, were known, since the 
masses of the proton and deuteron had both been determined by mass- 
spectroscopic means. 

Soon afterward Chadwick and Goldhaber (C8) observed that the 
deuteron can be decomposed into a neutron and a proton by the absorp- 
tion of a gamma-ray from ThC" (2.614 8 Mev). From the energy of the 
resulting proton they obtained a value of 2. 1 Mev for the deuteron bind- 
ing energy. Until recently the accepted neutron mass was obtained 
from accurate measurements of the threshold gamma-ray energy for 
the photodisintegration of the deuteron. 

However, quite recently Bell and Elliott (B16) have accurately 
measured the energy of the gamma-rays resulting from the capture of 
neutrons, of negligible kinetic energy, by protons (the reverse of the 
photodisintegration). Using their value of 2.230 ± 0.007 Mev for the 
gamma-ray energy, and other data, Bainbridge (Volume I, Part 5) 
gives the value M = 1.008982 atomic mass units for the neutron mass. 

3. Instability. With the first relatively accurate determination of the 
neutron mass (C8), which showed that it exceeds the mass of the hydro- 
gen atom, Chadwick and Goldhaber ventured the prediction that the 
neutron should be unstable against beta-decay, according to the reaction 
neutron — > proton + /3~ + neutrino. The maximum beta-ray kinetic 
energy for the decay of neutrons of negligible kinetic energy is given by 
the n — H mass difference, which we take to be equivalent to 782 ± 1 

1 That the neutron mass behaves in the conventional fashion with respect to gravi- 
tation has been directly verified by McReynolds (M19), who measured the "free 
fall" of slow neutrons. 



Sec. IB] Properties and Fundamental Interactions 



217 



kev (T16). 1 Assuming that the decay of the neutron follows the empiri- 
cal laws of beta-decay, being an allowed transition with a log ft value of 
~3.5, the half-life of the neutron should be ~20 min. 

Recent experiments in two laboratories have established that the 
neutron does decay, as per the above predictions. In the first ones, per- 
formed at the Oak Ridge National Laboratories by Snell, Miller, Pleason- 
ton, and McCord (S43, S44), the neutron decay has been observed 
through the simultaneous (coincidence) detection of the beta-particle 




200 300 400 BOO 
Energy (kev) 

Fig. 1. Fermi plot of the negatrons from the decay of neutrons, due to Robson 
(R15). The deviations from the allowed shape, below 300 kev, are instrumental. 

and proton, resulting from the decay of neutrons in an intense beam 
passing through an evacuated chamber containing the particle detec- 
tors. Control experiments were performed to establish that the coinci- 
dences occurred only in the presence of the neutron beam, and that the 
beta-particles were of roughly the expected energy. These experiments 
are consistent with "a half-life in the range 10-30 minutes." 

Robson (R15) at the Chalk River Atomic Energy Project, in Canada, 
has not only observed the neutron decay and measured its half-life but, 
in an experiment of exemplary care and ingenuity, has also succeeded in 
measuring the spectrum of the decay electrons. He first observed the 
positive heavy particles resulting from the decays (in a highly sensitive 
mass spectrometer) and established that they are protons. He then 

1 Actually, in the decay of a neutron at rest the emission of a ^-particle and a 
neutrino requires, for the conservation of momentum, that some energy be given to 
the proton. However, for the emission of a /3-ray of the maximum possible energy, 
the kinetic energy of the recoil proton is only 0.43 kev. 



218 The Neutron [Pt. VII 

succeeded in obtaining coincidences between the protons and their 
associated beta-particles, detected in a magnetic lens electron spectrom- 
eter in which he simultaneously measured the energy of the beta-parti- 
cles. The beta-spectrum obtained in this manner is shown in Fig. ] , a 
conventional allowed-transition Fermi plot. The end point corresponds 
to a maximum beta-ray kinetic energy of 782 ± 13 kev. From an (ex- 
perimental plus computational) evaluation of the geometrical efficiency 
of proton detection and a measurement of the neutron density in the 
beam (only ~10 4 neutrons/cm 3 ), Robson deduced the half-life of 12.8 
±2.5 min for the decay of the neutron. 

Other experiments have been suggested for the observation of the 
decay of the neutron and the measurement of its half-life. Among the 
most interesting is the following: A sealed, evacuated, thin- walled con- 
tainer of material with low neutron absorption, say glass, is placed in a 
region of high neutron density. After some time has elapsed, the 
accumulation of hydrogen gas in the container is measured. The con- 
tainer acts as a semi-permeable membrane, permitting the free entry of 
neutrons but preventing the escape of hydrogen formed by neutron 
decay in the container. For a neutron density of about 0.5 X 10 10 
neutrons/cm 3 (corresponding to a thermal neutron flux of ~10 15 
neutrons • cm" -2 • sec -1 , of average velocity ~2 X 10 s cm/sec) a hydrogen 
pressure of ~10~ 4 mm of Hg would be developed in the container in 
approximately one month ; such an accumulation of hydrogen gas should 
be easily observable. The difficulties of this experiment involve the 
attainment of sufficiently high neutron fluxes (see Section 3), possible 
production of hydrogen through (w,p) reactions in the walls of the con- 
tainer, and the necessity for complete removal of hydrogen from the 
container (walls) prior to irradiation. 

4. Spin and Statistics. Like the electron and the proton, the neutron 
has a spin of \ and obeys Fermi-Dirac statistics. The evidence for the 
spin value of \ is quite conclusive, although it involves a combination 
of experimental observations and the theoretical deductions therefrom. 
Some of these experiments involve the interaction between neutrons 
and protons and the properties of the deuteron; they will be further dis- 
cussed in Section 1C. 

The most important evidences for the value of \ for the spin of the 
neutron are: (1) the cross section and energy dependence of the scatter- 
ing of neutrons by protons, taken in conjunction with the evidence con- 
cerning the neutron-proton force, derived from the binding energy and 
spin of the deuteron; (2) the values of the magnetic moments of the 



Sec. IB] Properties and Fundamental Interactions 219 

deuteron, proton, and neutron; (3) the coherent scattering of neutrons 
by hydrogen, as evidenced in experiments involving the scattering by 
hydrogen-containing crystals, total reflection from hydrogen containing 
"mirrors," and the scattering by ortho- and para-hydrogen; (4) the 
polarization of neutrons by scattering in ferromagnetic materials and, 
in particular, by total reflection from magnetic mirrors. The results of 
these experiments, coupled with the success of the semi-empirical theory 
of nuclear forces in explaining the observed properties of the deuteron 
and many of the properties of light nuclei, constitute a most convincing 
argument for the correctness of the assignment to the neutron of a 
spin of J. 

As a particle of odd half-integral spin, the neutron is expected to 
obey Fermi-Dirac statistics. All the available evidence supports this 
expectation. The most important arguments concerning the statistics 
obeyed by the neutron involve (1) properties of the light nuclei — in 
particular, the fact that the deuteron and alpha-particle are known to 
obey Bose-Einstein statistics — which can be qualitatively understood 
only by use of the exclusion principle (in a fashion closely analogous to 
its application to the qualitative understanding of the periodic table), 
and (2) the saturation of nuclear forces, as manifested in the constant 
density of nuclear matter (the proportionality of the nuclear radius 
to A*). 

5. Magnetic Moment. 1 The first strong indication that the neutron 
has an intrinsic magnetic dipole moment came from the observation 
that the moments of the proton and deuteron are very different : nd < n P 
by <~2 nuclear magnetons. Since the deuteron is known to have spin 
I = 1, it is expected, on the basis of the simplest reasonable deuteron 
structure (a neutron and a proton in a 3 Si state), that m = y. p + \x n . 

1 The neutron, having spin \, cannot show any moments higher than a dipole 
moment. Although the possibility of the existence of an electric dipole moment of 
the neutron is usually dismissed, on the basis of theoretical arguments involving the 
concepts of symmetry and parity, it has recently been pointed out by Purcell and 
Ramsey (P32) that the assumptions on which these arguments are based are still not 
completely proved. Thus, for example, "if the nucleon should spend part of its time 
asymmetrically dissociated into opposite magnetic poles of the type that Dirac has 
shown to be theoretically possible, a circulation of these magnetic poles could give 
rise to an electric dipole moment." Although the detection of a possible electric 
dipole moment of the neutron is experimentally rather difficult, the above authors 
suggested that it could be observed in a modification of the experiment of Alvarez 
and Bloch (A13); they proposed to detect a shift of the neutron precession frequency, 
caused by the application of a strong electric field. However, such an experiment 
by Purcell, Ramsey, and Smith (S40) yielded a negative result and placed an upper 
limit, on the neutron's electric dipole moment, of two opposite electron charges sepa- 
rated by a distance of 5-10 -21 cm. 



220 The Neutron [Pt. VII 

The magnetic moment of the neutron has been measured by an in- 
genious modification of the Rabi-type molecular beam magnetic reso- 
nance experiment (Rl). In this experiment the usual polarizing and 
analyzing (focusing) inhomogeneous magnetic fields are replaced by 
slabs of magnetized iron whose property of preferentially transmitting 
neutrons of one direction of polarization is used in a manner completely 
analogous to an optical polarimeter. (The polarization of neutrons 
by transmission through magnetized iron, the Bloch effect, will be dis- 
cussed in Section 5D.) 

Early measurements of the neutron magnetic moment (F47, P28) 
gave a value of <~ — 2 nuclear magnetons. The negative sign, which was 
definitely established by observing the direction of precession of neutrons 
in a known magnetic field, means that the neutron angular momentum 
and its magnetic moment are oppositely directed. 

The first accurate measurement of the neutron moment, by Alvarez 
and Bloch (A13), yielded the value | n n \ = 1.935 ± 0.02 nuclear magne- 
tons. More recent experiments (A34, B39, R16) have measured, with 
high precision, the ratio of the neutron moment to the proton moment. 
These measurements, when combined with the latest value of the pro- 
ton moment, yield (Ml) 

fi n = -1.91280 ± 0.00009 nuclear magnetons 

6. Wave Properties. In accordance with the laws of quantum me- 
chanics we expect that the neutron should, under the appropriate experi- 
mental conditions, exhibit wave properties. Associated with neutrons 
of kinetic energy E (velocity v, momentum p) there is a wavelength 

x = 1 = Wev> (for " <<c) (1) 

Table 1 gives the neutron wavelengths corresponding to an assortment 
of neutron energies. The energy is expressed both in electron volts and 
in degrees Kelvin, the temperature T corresponding to a given energy 
E being defined by the relationship 

E = kT (2) 

where E is expressed in ergs and k is Boltzmann's constant, k = 1.3803 
X 10 -16 erg/°K. Also included in Table 1 is the "Dirac wavelength," 
X = X/2t, which is frequently the form in which the neutron wave- 
length enters into the expressions with which we shall be concerned. 
The classification of neutrons into types, according to their energy, will 
be described in Section 1B7. 



Sec. IB] Properties and Fundamental Interactions 

TABLE 1 
Wavelengths Associated with Various Neutron Energies 



221 



E(ev) 


T(°K) 


v (cm/sec) 


X (cm) 


X (cm) 


Type 


0.001 


11.6 


4.37 X 10 4 


9.04 X 10" 8 


1.44 X 10" 8 


Cold 


0.025 


290 


2.19 X 10 5 


1.81 X 10" 8 


2.88 X 10" 9 


Thermal 


1.0 


1.16 X 10 4 


1.38 X 10 6 


2.86 X 10 ~ 9 


4.55 X 10- 10 


Slow (reso- 
nance) 


100 


1.16 X 10 6 


1.38 X 10 7 


2.86 X 10" 10 


4.55 X 10- 11 


Slow 


10 4 


1.16 X 10 8 


1.38 X 10 8 


2.86 X 10" 11 


4.55 X 10" 12 


Intermediate 


10 6 


1.16 X 10 10 


1.38 X 10 9 


2.86 X 10- 12 


4.55 X 10 " 13 


Fast 


10 8 


1.16 X 10 12 


1.28 X 10 10 


2.79 X 10~ 13 


4.43 X 10- 14 


Ultrafast 


10 10 


1.16 X 10 14 


2.99 X 10 10 


1.14 X 10- 14 


1.81 X 10~ 16 


Ultrafast 
(relativis- 
tic) 



For the purpose of rapid calculation it is convenient to note the fol- 
lowing relationships: 

r(in °K) = 1.16 X 10 4 # 

Kin cm/sec) = 1.38 X 10 G E y ° 

X(in cm) = 2.86 X lO -9 ^ K 

X(in cm) = 4.55 X 10 -10 £ -H 

where E is always given in electron volts: 



1 ev = 1.602 X 10" 



ergs 



These expressions hold only in the non-relativistic energy region, i.e., 
for kinetic energies well below the energy corresponding to the neutron 
rest mass, 

Mc 2 = 939.5 million electron volts (Mev) 

As can be seen from Table 1, the relativistic effects (on the connection 
between v, or X, and E) are already evident, albeit still small, at a kinetic 
energy of 100 Mev. 

In general, the importance of the wave characteristics of the neutron 
is determined by the magnitude of the ratio between the neutron (Dirac) 
wavelength and the dimensions of the system with which the neutron 
is interacting. For neutrons of wavelength large compared to nuclear 



222 The Neutron [Pt. VII 

dimensions, the wave properties are of primary importance in deter- 
mining the nature of the interaction between neutrons and nuclei. 
Since nuclear radii fall in the range 2 X 10~ 13 to 10 -12 cm, the wave 
properties of the neutron are seen to be important for energies up to 
the fast neutron region; for fast neutrons, the wave properties are of 
comparatively lesser significance. 

The wavelengths of thermal neutrons are of the same order as the 
interatomic distances in solid matter. Thus, we would expect neutrons 
of these energies to show interference effects in their passage through, 
and scattering by, ordered materials. Such interference phenomena were 
predicted by Elsasser (E5) and by Wick (W24) in 1936-37, and soon 
indicated by Preiswerk and von Halban (P29) and by Mitchell and 
Powers (M33). Indeed, thermal neutrons have been found to behave 
very much like x-rays under similar circumstances, with the important 
difference that the scattering and absorption of neutrons are nuclear 
phenomena, whereas the corresponding properties of x-rays arise from 
their interaction with atomic electrons. Neutron diffraction and inter- 
ference phenomena will be discussed in detail in Section 5. 

It is worth observing that, although wave effects are relatively less 
important for fast neutrons, they may possibly become significant again 
for ultrafast neutrons. In this energy range the neutron wavelengths 
are of the order of the distance between nucleons in the nucleus, and 
the scattering from the nucleons within a nucleus can exhibit interfer- 
ence effects. Indeed, it is possible that, with the development of tech- 
niques for studying such effects, investigations of the "form factor" in 
nuclear scattering may provide important information on nuclear 
structure. 

7. Classification according to Energy. In the subsequent discussion 
we shall find it convenient to refer to neutrons of different kinetic ener- 
gies according to the following system of classification: 

I. Slow neutrons: < E < 1000 ev 

II. Intermediate neutrons : 1 kev < E < 500 kev 

III. Fast neutrons: 0.5 Mev < E < 10 Mev 

IV. Very fast neutrons: 10 Mev < E < 50 Mev 

V. Ultrafast neutrons: 50 Mev < E 

Although the dividing lines between these categories are quite arbi- 
trary, this system of classification can be justified on two grounds. In 
the first place, the interactions of the different classes of neutrons with 



Sec. IB] Properties and Fundamental Interactions 223 

nuclei and with matter in bulk involve, in general, different reactions 
and types of phenomena. Secondly, the methods of producing and de- 
tecting the different classes of neutrons are quite different. 

(a) Slow Neutrons. The behavior of neutrons in this energy range 
has been more extensively investigated than in any of the other ranges. 
The interaction of slow neutrons with heavy nuclei is characterized by 
sharp absorption resonances and large absorption cross sections for very 
low-energy neutrons. This is also the region in which crystal effects 
are important. 

Of particular importance to the study of the slow neutron region is 
the fact that a number of instruments, known as monochromators, have 
been developed whereby neutrons of a given energy may be singled out 
and their properties studied. Monochromators in the slow neutron re- 
gion are characterized by high resolving power, so that the dependence 
of a particular effect on the neutron energy can be ascertained to high 
accuracy. 

The slow neutron range is conveniently subdivided into a number of 
sub-ranges, of which the most important are : 

(1) Cold Neutrons. These constitute a special category of slow neu- 
trons of energy less than ^0.002 ev ; they exhibit an anomalously large 
penetrability through crystalline or polycrystalline materials. 

(2) Thermal Neutrons. In diffusing through materials with relatively 
small neutron absorption, slow neutrons tend to assume a velocity dis- 
tribution of the Maxwellian form 

dniv) = Av 2 e- M °* /2kT dv (3) 

The peak of the Maxwellian distribution is at an energy E — kT, where 
T is the absolute temperature of the medium through which the neutrons 
are diffusing. Neutrons having such a velocity distribution are referred 
to as thermal neutrons. 

Frequently the neutrons diffusing through a given medium are not 
in thermal equilibrium with the medium. This results in a rather 
greater preponderance of high-energy (slow) neutrons than is given by 
the Maxwell distribution function. Slow neutrons originating from such 
a source, generally of energy above ~0.5 ev, are often referred to as epi- 
thermal neutrons. 

In much of the work on slow neutrons the thermal neutrons are 
separated from neutrons of higher energy (say epithermal) by taking 
advantage of their strong absorption in relatively thin layers of cad- 
mium. Those neutrons in a given distribution which are absorbed by 



224 The Neutron [Pt. VII 

a cadmium layer are sometimes called C-neutrons; the neutrons which 
penetrate cadmium (energy > 0.3-0.5 ev) are, correspondingly, called 
epicadmium neutrons. 

(3) Resonance Neutrons. This classification refers to slow neutrons 
of energy between ~1 and ~100 ev; it is based on the large number of 
distinct, sharp, absorption resonances which have been observed in the 
interaction of neutrons, in this energy range, with heavy nuclei. 

(b) Intermediate Neutrons. In the intermediate range the predomi- 
nant type of neutron reaction is elastic scattering. Until fairly recently 
this region has been the least extensively studied, mainly because of the 
lack of suitable neutron sources and detectors. Recently a number of 
techniques have been developed for the study of this energy range. 
These techniques are now being extensively exploited. 

(c) Fast Neutrons. The fast-neutron region is characterized by the 
appearance of many nuclear reactions which are energetically impossible 
at lower neutron energies, of which the most important is inelastic scat- 
tering. This region has been fairly extensively investigated, although 
the available techniques have been rather crude as compared to those 
used in the slow-neutron range. 

(d) Very Fast Neutrons. This energy interval is distinguished from 
the preceding by the appearance of nuclear reactions involving the emis- 
sion of more than one product, such as the (n,2n) reaction. It is a rela- 
tively unexplored energy range, mainly because of the comparatively 
small number of suitable neutron sources. 

(e) Ultrafast Neutrons. The development of ultrahigh energy particle 
accelerators has resulted in the possibility of producing ultrafast neutrons 
and studying their properties. Before the advent of these machines, this 
region was accessible only through the utilization of the neutrons in the 
cosmic radiation. A distinguishing feature of nuclear reactions in this 
energy range comes from the relatively small interactions of neutrons 
with nuclei, resulting in a partial transparency of nuclei to ultrafast 
neutrons. Also, for these high energies, "spallation reactions" — in 
which the bombarded nucleus emits many fragments — are observed. 

C. Fundamental Interactions 

Of primary importance for the understanding of nuclear forces is the 
study of the interaction between neutrons and other nucleons — neutrons 
and protons — and between neutrons and the lighter particles (electrons, 
mesons) . There have been many experimental and theoretical investiga- 
tions which shed light on the nature of nuclear forces. Among the most 
important are the scattering of neutrons (and protons) by protons, the 



Sec. 1C] Properties and Fundamental Interactions 225 

capture of neutrons by protons, the static and dynamic properties of 
the deuteron, the scattering of neutrons by hydrogen molecules, and 
some of the properties of heavy nuclei. Since most of these results have 
already been discussed in Part IV of Volume I, they will merely be 
presented here in summary form. 

However, since a number of aspects of the neutron-proton interaction 
are of primary importance for the understanding of the interaction of 
neutrons with matter, such aspects will be discussed more fully. Thus, 
the scattering and capture of neutrons by protons will be covered in 
some detail. The discussion of the scattering of neutrons by hydrogen 
molecules, involving as it does the wave properties of neutrons, will be 
reserved for Section 5. 

1. The Neutron-Proton Interaction, (a) Properties of the Deuteron. 
(1) Static Properties. The ground state of the deuteron is essentially a 
3 S% configuration. 1 From the known binding energy of the deuteron, 
conclusions can be drawn concerning the force (potential) between the 
neutron and proton in this state, in which the neutron and proton spins 
are aligned in the same direction (total spin = spin of the deuteron 
= 1). The static properties of the deuteron give no information con- 
cerning the neutron-proton potential in the singlet state (total spin = 0) 
(B22). 

(2) Photodisintegration. The deuteron can be disintegrated into a 
neutron and a proton by the absorption of an amount of energy greater 
than its binding energy, 2.23 Mev. Although this can be accomplished 
through the bombardment of deuterons by a variety of particles, the 
disintegration of the deuteron was first observed (C8) in bombardment 
by gamma radiation. This process, the so-called photodisintegration 
of the deuteron, has been most extensively investigated, and has yielded 
much useful information concerning the neutron-proton interaction 
(B22). 

Immediately after its discovery, Bethe and Peierls (B20) and Massey 
and Mohr (M6) propounded the theory of the disintegration of the 
deuteron by a photoelectric effect. Soon afterward, Fermi (F15) sug- 
gested a second mechanism which contributes to the disintegration: a 
photomagnetic process. In the photoelectric disintegration the electric 
field of the gamma-radiation acts on the instantaneous electric dipole 
moment of the deuteron (the average electric dipole moment is, in the 

1 The deuteron has a small electric quadrupole moment, which fact implies that 
the ground-state wave function contains, in addition to the predominant 3 Si func- 
tion, a small admixture of 3 Z>i form. The implications of this fact, especially with 
regard to the tensor nature of nuclear forces, has been discussed in detail in Part IV 
of Volume I. 



226 The Neutron [Pt. VII 

ground state, zero). After separation of the proton from the neutron, 
through the action of the electric field, the spins of the neutron and 
proton remain parallel; thus, the photoelectric effect involves only the 
properties of the neutron-proton interaction in the triplet state. In the 
second process the magnetic field of the radiation interacts with the 
magnetic dipole moments of the neutron and proton, the effect of the 
interaction being to "flip" the spin of one of the particles with respect 
to that of the second; thus, the photomagnetic process also involves the 
properties of the neutron-proton potential in the singlet state. 

The contributions of the two processes can be separated experimen- 
tally, since they result in different angular distributions of the emerging 
particles with respect to the direction of the incident gamma-ray. These 
angular distributions can be understood in terms of a rough, semi-classi- 
cal model of the deuteron as a neutron and proton separated by a fixed 
distance and having equal probability for all possible orientations of the 
connecting line. Thus, in the case of photoelectric disintegration, the 
effect is greatest when the deuteron's dipole moment is in the direction 
of the electric field vector of the gamma-ray beam, which is perpendicu- 
lar to the direction of motion of the photon. This leads to a distribu- 
tion of recoil neutrons and protons (in the center-of-mass coordinate 
system) proportional to the square of the sine of the angle between the 
incident photon and the recoils. 

In the photomagnetic disintegration, on the other hand, the effect is 
produced by the difference between the action of the photon's magnetic 
field on the neutron and the proton, which is due to the difference be- 
tween the magnetic dipole moments of the two particles [—1.91280 and 
+2.79255 nuclear magnetons, respectively (Ml)]. This difference is 
independent of the orientation of the deuteron, and hence the recoils, 
which are due to the photomagnetic disintegration, are spherically sym- 
metrically distributed in the center-of-mass coordinate system. 1 

In addition to the different angular distributions, the cross sections 
for the photoelectric and photomagnetic disintegrations have different 
dependences on the gamma-ray energy. The cross sections start out at 
zero at the photodisintegration threshold, rise to maxima for values of 

1 This difference in the angular distributions of the recoil products follows from 
simple quantum-mechanical considerations. Thus, in the photoelectric effect the 
final state of the separated neutron and proton must be a P state to satisfy the 
selection rule for electric dipole radiation, Ah = ±1; also, since the electric field 
does not act, in first order, on the nucleon spins, AS - 0. The transitions occur 
from the state of the deuteron for which m = ±1 (with respect to the direction of 
the incident photon); hence the sin 2 distribution of the recoils. Correspondingly, 
the magnetic dipole transition requires AL = 0, AS = ±1. Thus, the final state is 
a 1 S a state, with a spherically symmetrical distribution of the recoils. 



Sec. 1C] Properties and Fundamental Interactions 227 

E y = — 2.2 Mev of approximately twice the binding energies (of the trip- 
let and singlet states, respectively), and then fall off with increasing 
gamma-ray energy. 1- For photon energies close to the threshold, the 
two effects are comparable in importance. For photon energies large 
compared to the binding energy of the deuteron, the photomagnetic 
effect is negligible compared to the photoelectric effect. Thus, the 
angular distribution of the recoils can be described by the expression 
a + b sin 2 0, with a and b both energy-dependent, and a falling off much 
more rapidly than b with increasing gamma-ray energy. For photon 
energies considerably above the threshold, the simple considerations 
described above are no longer completely applicable; other effects, such 
as the tensor nature of the neutron-proton interaction, come into 
play (R6). 

Since the early attempts to observe the cross section and angular 
distribution of the recoils in the photodisintegration of the deuteron 
(CIO), the techniques of measurement have been considerably improved. 
More recent results (B30, G28, W44) confirm the theoretical predictions 
and provide important information concerning the nature of the neutron- 
proton interaction. 

(6) The Capture of Neutrons by Protons. Although, for neutrons of 
most energies, the most important effect in the neutron-proton interac- 
tion is the scattering process, discussed in (c) below, it was soon observed 
(W18, D19, A20) that neutrons are appreciably absorbed in hydrogen- 
containing substances. This absorption is attributable to a radiative 
capture (n,y) process, a reaction to be expected, since the combination 
of a neutron and a proton into a deuteron is energetically preferred. 
The gamma-rays accompanying this process were first observed by Lea 
(L10). The radiative capture reaction, n + p — > d + 7, is the inverse 
of the photodisintegration of the deuteron, and the calculation of its 
cross section follows directly from the photodisintegration calculations 
(B22). 

Such a calculation indicates that capture of a slow neutron by a photo- 
electric process (emission of an electric dipole gamma-ray) has a negligi- 
ble cross section (B20) ; this result is easily understood, since photoelec- 

1 If (3 = 2.2 Mev and | n | = 0.065 Mev are the absolute values of the binding 
energies in the triplet and singlet states (the singlet state is actually unbound) and 
E = E y — €3, then the simple theory gives (B22) 

&re 2 h 2 t3 A E* 



ff mag — 



3 he M (E + e 3 ) s 
2ir e 2 h 2 Qfr - ;Q 2 («* + 1 q |*)»«,*g* 

3 he M Mc 2 (E + caKE + |ta I) 



228 The Neutron [Pt. VII 

trie capture requires that the neutron-proton system be, initially, in a 
P state, which is highly improbable for a slow neutron. Indeed, it was 
the observation of an appreciable neutron-proton capture cross section 
which led Fermi (F15) to postulate a photomagnetic capture process 
and its inverse, the photomagnetic disintegration. Since the photo- 
magnetic capture involves only S states C-S —> s Si), the resulting 
cross section has al/» dependence on the neutron energy and therefore 
becomes quite appreciable for thermal neutrons. Fermi showed that 
the photomagnetic capture is indeed strong enough to account for the 
relatively rapid absorption of thermal neutrons in hydrogenous materials. 

The precise experimental determination of the neutron-proton (n,y) 
cross section is difficult, since capture represents only a small fraction 
of the total neutron-proton cross section at even the smallest available 
neutron energies and, further, it leads to a non-radioactive end product. 
Thus, none of the usual methods of measuring cross sections (Section 3) 
is easily applicable. Its value can, however, be inferred from measure- 
ments involving the diffusion of thermal neutrons in hydrogen-contain- 
ing materials (Section 4), since the average distance traveled, before 
absorption, by a thermal neutron depends on the capture as well as on 
the scattering cross section (A20). Or, equivalently, the mean life of a 
thermal neutron in a hydrogenous material is inversely proportional to 
the capture cross section, so that a measurement of the mean life can 
yield a value for <r(n,y) (A18, M4). 

The most accurate value of the (n,y) cross section has been obtained 
from a direct comparison of slow neutron-proton absorption with the 
absorption of boron, which also has a 1/v energy dependence (R8, W19). 
Whitehouse and Graham (W19) obtained 

— = 2270 ± 68 

Using the value <tb = 710 ± 21 barns for neutrons of velocity v = 
2200 m/sec (R20), this yields 

<m(n,y) = 0.313 ± 0.013 barns 

Since the photomagnetic capture and disintegration processes involve 
the properties of the neutron-proton interaction in the singlet state, the 
magnitudes of the cross sections can be used to infer the strength of this 
interaction. In particular, different cross-section values are predicted, 
depending on whether the singlet state is bound or unbound (B22). In 
the case of the photodisintegration experiments, the cross-section deter- 
minations are not sufficiently accurate to allow an unambiguous choice 
between the two possibilities (G28). 



Sec. 1C] Properties and Fundamental Interactions 



229 



The neutron-proton radiative capture cross section is measured with 
sufficient accuracy. Rosenfeld (R19) calculates the cross section, for 
neutrons of energy 0.026 ev, to be 0.32 or 0.16 barn for the cases, respec- 
tively, of an unbound or a bound singlet state. The experiments 
strongly favor the conclusion that the ^o state of the deuteron is un- 
bound. "But [chiefly because of the neglect of the exchange effect] the 
calculation is not sufficiently accurate to allow us to regard this evidence 
as entirely conclusive." 

The conclusion is, however, completely borne out by the measure- 
ments on the scattering cross sections for ortho- and parahydrogen (see 
Section 5). 

(c) The Scattering of Neutrons by Protons. (1) Cross Section. Since 
the first investigations of Chadwick (C7), there have been many meas- 
urements, at various neutron energies, of the cross section for the scat- 
tering of neutrons by protons. The results are summarized in Fig. 2, 



_, 2 

M 

a 

u 

t S 
b 6 

4 























































H 





















































































































































































































































































































































































































































































































































































































































































































































































































































































































0.03 

23468 23468 28468 23468 2 

0.01 0.1 1 10 100 300 

E n (Mev) 

Fig. 2. Cross section for the scattering of neutrons by free protons vs. kinetic energy 

of the neutron, in the range 0.01-300 Mev, from the compilation of Adair (A2). 

Additional data, in the ultrafast range, are given in (T3) and (D12). 



in which the neutron-proton scattering cross section is plotted as a func- 
tion of the neutron energy, between 0.01 and 300 Mev. 

The quantum-mechanical treatment of neutron-proton scattering was 
first given by Wigner (W27), who showed that the cross section as a 



230 The Neutron [Pt. VII 

function of neutron kinetic energy E is given (in the limit of neutron 

wavelength large compared to the range of the neutron-proton force) by 

the expression „ 

4irh 2 

er(n-p) = ■ = : — j- (4) 

M(^+|c|) 

where e is the binding energy of the neutron-proton system (deuteron). 
The above expression must be corrected for the finite range of the neu- 
tron-proton force (B22); however, this correction is independent of E 
and need not be considered for the purposes of the arguments which 
follow. The above expression was found to agree with the observed 
cross sections for neutrons of ~1 Mev and greater. 

When applied to slow neutrons, the above expression yields a value 
of ~3 barns. The early experiments (D19) gave a value, very much 
greater, of ~35 barns. (The cross section for the scattering of neutrons 
by free protons has, in fact, been determined to be ~20 barns.) 

This serious discrepancy was explained by Wigner x under the assump- 
tion that the force between a neutron and proton depends on the rela- 
tive orientation of their spins. Since in a fourth of the cases the neutron- 
proton scattering takes place in the singlet state, the cross section should 
be given by the expression 

47rh 2 /3 1 1 1 \ 

c(n-p) = 1 - i r -\ — ^ i r ) (5) 

M \4|^+|6 3 | 4|tf+U|/ 

where e 3 and « x refer, respectively, to the binding energies of the triplet 
and singlet states of the deuteron. To fit a slow-neutron cross section 
of 20 barns, | ei | = 0.065 Mev. 

From the neutron-proton scattering, it is impossible to determine the 
sign of €i. Actually, we know from other evidence that the singlet state 
of the deuteron is unbound. Thus, although the term "binding energy" 
has no meaning for this state, «i represents a certain combination of the 
constants which describe the neutron-proton interaction in the singlet 
state (B22). 

Early measurements of the slow neutron-proton scattering cross 
section (D19, W18, A20) showed a wide variation, outside the experi- 
mental uncertainties. The reason for these variations was given by 
Fermi (F17), who ascribed them to the effect of the binding of the pro- 

1 Although it is universally acknowledged throughout the literature (F5, B22) that 
Wigner originated the idea of the spin dependence of the neutron-proton force to 
explain the large neutron-proton scattering cross section for slow neutrons, he has 
not, to our knowledge, published these considerations. 



Sec. 1C] Properties and Fundamental Interactions 231 

tons in the material — molecule, liquid, or solid — used as a neutron scat- 
tered Such effects are, of course, negligible for neutron energies con- 
siderably greater than the binding energy of the proton in the system 
under consideration. They are most important for neutron energies 
less than the lowest excitation energy of the system in which the proton is 
bound. The proton behaves like a particle of infinite mass when bound 
in solids or liquids, or like a particle of the total mass of the molecule 
involved in the case of neutron scattering by gaseous materials. It can 
be shown that the scattering cross section of slow neutrons by bound 
protons varies directly as the square of the reduced mass of the neutron, 
the scattering being spherically symmetrical in the center-of-mass co- 
ordinate system. Thus, the scattering cross section of infinitely slow 
(zero energy) neutrons by protons in solids or liquids should approach 
four times the free proton cross section, or about 80 barns. 

The variation of the neutron-proton cross section in the energy range 
in which the proton is neither free nor completely bound (energy greater 
than the lowest excitation energy but less than the binding energy of the 
protons) depends on the specific properties of the proton-containing 
material under consideration and is rather difficult to compute. Dis- 
regarding interference effects due to the crystalline properties of the 
solid under consideration or to the order introduced by the molecular 
structure (see Section 5), the cross section can be shown to decrease 
monotonically from the bound proton to the free proton cross section 
as the neutron energy increases from to several electron volts (A31, 
M27). 

Thus, the free proton scattering cross section for slow neutrons can 
be obtained from measurements on neutrons in the resonance energy 
range. As a result of such measurements (C18), the slow neutron-free 
proton scattering cross section was determined to be slightly less than 
21 barns. However, binding effects still have a small influence for neu- 
trons with energy of a few electron volts. In the most recent experi- 
ments (M24), the neutron-proton cross section has been measured as a 
function of the neutron energy, and theoretical considerations (P19) 
were used to extrapolate to a free proton neutron-proton scattering 
cross section of 20.36 ± 0.10 barns for neutrons of zero kinetic energy. 

Effects of nuclear binding are, of course, present in the scattering of 
neutrons by heavier nuclei, although their importance decreases rapidly 
with increasing mass number A of the scattering nucleus. In general, 
for nuclei bound in solids or liquids, 

(A + 1\ 2 

Cbound — I I ""free (6) 



232 



The Neutron 



[Pt. VII 



Thus, for deuterons bound in solid material, <rb OU nd = 2.25<rf re e, while, 
for the scattering of slow neutrons by carbon, abound = 1.17o-f ree . 

(2) Angular Distribution. It was first pointed out by Wigner (W27) 
and by Wick (W22) that the angular distribution, in the scattering of 



20 - 



-15 - 



slO 







1 1 1 1 




1 


1 ' ' 1 

o 


"""»>. 






„ *■"' 


V N 






°s' 


v s 40 Mev 






s 
s 


\ 






V 


\ 






/ 


\ 






/ 


\ 






/ 


\ 






fa ° 


\ 






' T 


\ 






• 


v 






/ 


— s 






s 


° o 


o" 


-"-'o' 


s'o ° 


=~^ n 90 Mev 
s 






-A ' 


N T 

- — tk~ 






\ / \- 
.7- V 

y 

j. 

1 1 1 


\260 *Xi 
\Mev £4rj- T 
- \ S^.x 

h ■■ 

1 1 1 1 


5 


.y 

i 



20 



40 



60 80 100 120 140 160 180 

Scattering angle 

Fig. 3. Angular distributions in the scattering of ultrafast neutrons by protons. 
The figure is from a paper by Kelly, Leith, Segre, and Wiegand (K9). 

fast neutrons by protons, should provide information concerning the 
neutron-proton interaction in states of higher angular momentum (P, D, 
etc). The scattering of slow neutrons by protons involves only the 
state of zero angular momentum (S state) and is therefore spherically 
symmetrical in the center-of-mass coordinate system. The energies at 
which higher angular momentum scatterings set in measure the strengths 



Sec. 1C] Properties and Fundamental Interactions 233 

of the interaction in these states, while the form of the angular distribu- 
tion leads to inferences concerning the nature of the potential. 

A number of experiments using d + d neutrons (2.4 to 2.7 Mev) show 
that, at these energies, the neutron-proton scattering is spherically sym- 
metrical in the center-of-mass system (R19). While the scattering is 
still essentially spherically symmetrical at 14 Mev (BIO), some of the 
experiments in the 9-14-Mev neutron energy range (L7, P24) show indi- 
cations of a slight asymmetry, corresponding to a preferential scattering 
of protons in the forward direstion. This asymmetry is quite pro- 
nounced for neutrons of energy 27 Mev (B71). Such an effect is to be 
expected on the basis of a theory of a neutron-proton interaction of the 
exchange type; exchange forces are required to explain the "saturation" 
of nuclear forces, since they lead to a repulsive force at short distances 
and thus tend to prevent the collapse of heavy nuclei and to maintain 
a constant density of nuclear matter. 

Neutron-proton scattering experiments with 40-, 90-, and 260-Mev 
neutrons show large deviations from spherical symmetry; their results 
are plotted in Fig. 3. The most interesting characteristic of these re- 
sults is the rough symmetry about 90° (in the cm. system) of the dif- 
ferential neutron-proton scattering cross section. Serber has pointed 
out that these results indicate a neutron-proton interaction involving 
approximately equal proportions of forces of the ordinary and exchange 
type. 

Christian and Hart (CI 5) have discussed the type of neutron-proton 
interaction which is required to fit both the low-energy and the high- 
energy data. They use an interaction potential of the Serber type, 
choosing the radial dependence to obtain the best possible fit, and also 
taking into account the tensor form of the interaction, required by the 
electric quadrupole moment of the deuteron. 

The Serber interaction, a mixture of equal parts ordinary and exchange 
forces, has the property that all terms which correspond to odd orbital 
angular momentum quantum numbers vanish; the observed symmetry 
of the high-energy neutron-proton scattering about 90° is, of course, 
the decisive evidence in favor of the Serber interaction, since any appre- 
ciable interaction in a state of odd orbital angular momentum would 
destroy this symmetry. However, the terms of odd angular momentum 
are precisely those which lead to a strong repulsion between the neutron 
and proton for small distance of separation and, therefore, are required 
for the "saturation" of nuclear forces. Indeed, it appears doubtful 
whether a nucleon-nucleon interaction of the Serber type is consistent 
with the saturation of nuclear forces in heavy nuclei. 



234 The Neutron [Pt. VII 

2. The Neutron-Neutron Interaction, (a) Experimental Evidence. 
(1) Low Energies. Since the highest available neutron densities are 
still far from appreciable (below 10 10 neutrons/cm 3 ), it is manifestly im- 
possible to obtain information about the neutron-neutron interaction 
through observation of the scattering of neutrons by free neutrons; our 
knowledge of the force between two neutrons must, perforce, be inferred 
from other information. In considering the available information, it is 
useful to bear in mind the following general considerations: (1) Experi- 
ments involving neutrons of kinetic energy below ~20 Mev yield infor- 
mation only about the neutron-neutron forces in the singlet S state since, 
at these energies, states of higher angular momentum are not appreciably 
excited, and the exclusion principle prevents the neutron-neutron system 
from existing in a triplet S state. Experiments with ultrafast neutrons 
can, on the other hand, yield information about the neutron-neutron 
interaction in both the singlet and triplet states. The same arguments 
hold, of course, for the proton-proton interaction. (2) Since there is 
some reason to expect that the purely nuclear interaction between two 
nucleons should be independent of whether they are neutrons or pro- 
tons (see further on), the evidence on the proton-proton interaction can 
be considered to have some bearing on the problem of the neutron- 
neutron interaction. In any event, we shall herein review, briefly, the 
available information on the interaction between two nucleons, irrespec- 
tive of their charge. 1 

The strongest evidence concerning the charge symmetry of nuclear 
forces (equality of the nuclear neutron-neutron and proton-proton 
forces) is derived from the observed energy differences between mirror- 
image nuclei (nuclei which can be obtained from one another by inter- 
changing the neutrons and protons). The energy differences between 
such nuclei, as measured by the maximum energy of the beta-rays 
emitted in the decay of one of the pair into the other, can be completely 
accounted for by the neutron-proton mass difference and the difference 
in the electrostatic energies of the two nuclei. In addition, mirror nuclei 
are similar with respect to their observed level structure. From this 
evidence, we conclude that the purely nuclear neutron-neutron and 
proton-proton forces are essentially equal. Actually the evidence on 
nuclear level structure lends strong support to the stronger hypothesis 
of charge independence (equality of the neutron-neutron, neutron- 
proton, and proton-proton forces in the same states), since nuclear iso- 
bars are observed to exhibit markedly similar level patterns. 

1 A much more complete discussion of these points is contained in Part IV of Vol- 
ume I, 



Sec. 1C] Properties and Fundamental Interactions 235 

The purely practical restrictions, which prevent the performance of 
neutron-neutron scattering experiments, do not obtain in the case of 
proton-proton scattering. Many such experiments have been performed 
for proton energies up to <~15 Mev. From these it can be concluded 
that within the experimental uncertainty (and taking into account the 
effect of the Coulomb interaction) the nuclear neutron-proton and pro- 
ton-proton forces in the 1 S state appear to be equal. 

Thus, the available evidence supports the idea of the charge inde- 
pendence of the nuclear forces between two nucleons. 

(2) On the Possible Existence of a Stable Di-Neutron. The strength 
of the neutron-neutron interaction in the '(So state determines whether 
or not there can be a stable system consisting of two neutrons. (There 
is no possibility for a stable two-nucleon state of higher orbital angular 
momentum.) The existence of such a stable "di-neutron" is, until now, 
experimentally neither proved nor disproved. If the nucleon-nucleon 
interaction is completely charge independent, the di-neutron will be 
unstable, since the '(So neutron-proton system is unstable by about 
65 kev; however, this is so close to being stable that it would require 
only a rather small increase of the neutron-neutron interaction over the 
corresponding neutron-proton interaction to lead to a stable di-neutron. 

A stronger argument against the existence of a stable di-neutron can 
be derived from the equality of the low-energy neutron-neutron and 
proton-proton forces, as evidenced by the properties of the mirror-image 
nuclei. Analysis of low-energy proton-proton scattering (Jl) proves, 
conclusively, that the '£ proton-proton interaction is not strong enough 
to lead to a bound state (even in the absence of the Coulomb repulsion). 

Although the weight of available evidence does not favor the existence 
of a stable di-neutron, it is, nevertheless, of interest to consider some of 
the possible consequences of its existence. Feather (F4) has pointed 
out that it is possible to obtain an upper limit to the binding energy of 
the di-neutron from the observation that the nucleus He 6 does not decay 
according to the reaction He 6 -»• He 4 + n 2 . From the masses of the 
nuclei involved, he concludes that the binding energy of the di-neutron 
is less than 0.7 ± 0.2 Mev. He also points out that the di-neutron 
would be beta-unstable, n 2 -» H 2 + P~ + v, with a mean life (assum- 
ing an allowed transition) of 1 < t < 5 sec. 

If the di-neutron should exist as a stable configuration, and if it should 
be possible to obtain it in sufficient numbers, it could easily be detected 
in experiments involving scattering by nuclei. For the two neutrons 
would be scattered coherently, and the interference effects would per- 
sist to considerably higher energies than do the normal coherence effects 
in slow-neutron scattering, since the di-neutron would be a compact sys- 



236 The Neutron [Pt. VII 

tem, of dimensions ~10 -12 cm. Thus, the scattering of di-neutrons by 
hydrogen atoms would be the same as the scattering of cold neutrons by 
parahydrogen molecules (see Section 5), for which the cross section is 
~4 barns, as compared to ~20 barns for the free neutron-proton scat- 
tering cross section. 

Furthermore, as pointed out by Feather, capture of a di-neutron 
would, as compared to ordinary neutron capture, result in a different 
compound nucleus and in characteristic radioactivities. 

The difficulty in observing such effects would, if the di-neutron were 
stable, arise from the difficulty of obtaining an appreciable source of 
di-neutrons. Such a source would have to be obtained from a suitable 
«.uclear reaction, in which di-neutrons are emitted. It could not be ob- 
tained by neutron-neutron collisions in a region of high slow-neutron 
density (even if neutron densities of sufficient magnitude were avail- 
able) because the required reaction, n + n — > n 2 + 7, would be highly 
forbidden, since it would involve the transition 1 S — * 1 £>o- Although 
a di-neutron could, if stable, be produced in a collision involving three 
neutrons, such collisions would be exceedingly improbable. 

Fenning and Holt (Fll) have attempted to detect the presence of 
di-neutrons in the Harwell pile, by looking for alpha-particles from the 

Bi 209 + n 2 -> AcC 211 — ^ AcC" -?-> Pb 207 

2.16 min 4.8 min 

(They were investigating the possibility that di-neutrons might be emit- 
ted in slow-neutron fission.) They exposed bismuth to a flux of ~10 12 
neutrons -cm -2 -sec -1 , but could detect no activity attributable to 
AcC, from which they established an upper limit of 1.5 X 10 -21 sec -1 
for the product of the di-neutron flux and the cross section for its absorp- 
tion in bismuth. 

In considering the possible effects of the di-neutron on nuclear reac- 
tions, it should be noted that such effects might exist even if the di- 
neutron is not stable, for the dynamics of a reaction involving two neu- 
trons would be quite different if the two neutrons were absorbed or 
emitted as a single unit than if they behaved independently (C19). 
Thus, Kundu and Pool (K25) consider the characteristics of the excita- 
tion of (H 3 ,p) reactions on rhodium and cobalt as "probable evidence of 
the di-neutron." Another, and more favorable, reaction in which effects 
of a di-neutron might be observed involves the two possibilities 

H 3 + H 3 -» He 4 + 2n + Q t 

-> He 4 + n 2 + Q 2 

If the two neutrons are emitted independently, their energies, and that 



Sec. 1C] Properties and Fundamental Interactions 237 

of the recoil alpha-particle, will vary over a rather wide range (for a 
given energy of the bombarding triton), as is to be expected in a three- 
particle reaction. If, on the other hand, the two neutrons are emitted 
as a single particle, the energies of the di-neutron and of the alpha- 
particle are uniquely determined, for a given angle of emission, by the 
laws of conservation of energy and momentum. The weight of available 
evidence favors the first of the two above-mentioned reactions (L22, 
A9, L14). 

Perhaps the most striking evidence concerning the di-neutron is 
derived as a by-product of the experiments of Panofsky, Aamodt, and 
Hadley (P3) on the absorption of negative pi-mesons by deuterium in 
the reaction ir~ + d — » In + y. In interpreting the observed gamma- 
ray energy distribution, Watson and Stuart (W6) point out that the 
shape is strongly dependent on the degree of correlation in the direc- 
tions of emission of the two neutrons; indeed, if the neutrons were always 
emitted as a di-neutron, the gamma-ray spectrum would be monochro- 
matic. The observations are sufficiently accurate to show that there is 
a strong interaction (attraction) between the two neutrons; they are not 
sufficiently precise, as yet, to allow more than an upper limit of <~200 
kev to be placed on the binding energy of the di-neutron. The proper- 
ties of the meson-producing reaction, 7 + d — » 2n + x + , might also 
throw some light on the di-neutron. 

(3) High Energies. For neutrons with kinetic energies in the ultra- 
fast range, it becomes possible to observe the neutron-neutron interac- 
tion somewhat more directly in the scattering of neutrons by deuterium 
nuclei. If the wavelength of the neutron is much less than the average 
separation between the neutron and proton in the deuteron, the two 
nucleons should scatter neutrons independently; the cross section for 
the scattering of neutrons by neutrons, a(n-n), should then be given by 
the difference between the scattering cross section of deuterium and that 

of hydrogen, ,_ 

e{n-n) = <j(n-d) — <r(n-p) V) 

Unfortunately, there are still appreciable effects of the structure of 
the deuteron in the scattering of neutrons of a few hundred Mev, since 
the neutron wavelength is not negligible compared to the internucleon 
distance. Another way of expressing this difficulty is to note that the 
nucleons in the deuteron are moving with kinetic energies of internal 
motion as high as <~25 Mev; the velocity, corresponding to this kinetic 
energy, is only ~i of the velocity of a 100-Mev neutron. 

A possible way out of this difficulty has been suggested by Segre. 1 
Since it is feasible to measure, directly, both a(n-p) and <r(p-p), a meas- 

1 Private communication. 



238 The Neutron [Pt. VII 

urement of o-(p-rf) should determine the magnitude of the effect, on the 
cross section, of the structure of the deuteron. This correction could 
then be applied to the measurement of o(n-d) in order to extract the 
neutron-neutron cross section. 

The interpretation of neutron-deuteron scattering for ultrafast neu- 
trons has been considered by a number of authors, e.g., De Hoffman 
(D8), Gluckstern and Bethe (G9), and Chew (C14). It turns out that, 
in addition to the expected contributions from neutron-proton and 
neutron-neutron scattering (both averaged over the momentum dis- 
tributions of the nucleons in the deuteron), there is a cross term due to 
interference between the two scattering centers and depending in de- 
tail on the nature of the interactions. As a result, the simple additivity 
relationship of Eq. (7) cannot even be applied at energies of a few hun- 
dred Mev. 

The most extensive attempt at interpretation of neutron-deuteron and 
proton-deuteron scattering experiments in the ultrafast energy range 
has been carried out by Chew (C14). He has treated the problem by 
the "impulse approximation," in which the scattering nucleon is as- 
sumed to behave in a relatively independent fashion during the time of 
the impact. This approximation appears to be appropriate for the 
treatment of some aspects of the neutron-deuteron scattering problem, 
e.g., the dissociation, by neutron impact, of the deuteron; however, 
other aspects, such as elastic neutron scattering or proton "pick-up," 
cannot be treated in this fashion. From the available data, Chew con- 
cludes that there is "no evidence for a difference between neutron- 
neutron and proton-proton interactions." 

The results of measurements, on the cross sections for scattering of 
ultrafast neutrons by protons and deuterons, are summarized in Table 2. 1 

TABLE 2 
Total Cross Sections of Protons and Deuterons for Ultrafast Neutrons 

cr(n-d) <rin-p) Difference 

E n (Mev) (barns) (barns) (barns) Reference 

42 0.289 ±0.013 0.203 ± 0.007 0.086 ±0.015 H61 

85 0.117 ±0.005 0.083 ±0.004 0.034 ±0.006 C25 

95 0.104 ±0.004 0.073 ±0.002 0.031 ± 0.004 D9 

270 0.057 ±0.003 0.038 ± 0.002 0.019 ±0.003 D10 

280 0.049 ±0.005 0.033 ±0.003 0.016 ± 0.006 F41 

(b) Some Comments Pertaining to the Meson Theory of Nuclear Forces. 
On the basis of the observed properties of the forces between two nu- 
cleons (in particular, the short range), Yukawa predicted the existence 

1 For results of p-p scattering measurements see Volume I, Part IV, Section 2A. 



Sec. 1C] Properties and Fundamental Interactions 239 

of a particle of mass intermediate between that of the electron and the 
proton. According to his idea the force between two nucleons is regarded 
as resulting from the interchange of such particles, called mesons, be- 
tween nucleons, in much the same way that purely electrical forces may 
be regarded as resulting from the interchange of photons between 
charged particles. There have been developed, since Yukawa's sugges- 
tion, a number of different types of meson theories of nuclear forces; 
these differ in the properties of the assumed mesons and in the type and 
strength of the assumed coupling between the meson and the nucleon 
fields (P7). 

Yukawa's prediction was soon followed by the discovery of such a 
particle in the cosmic radiation. This particle, the mu-meson or muon, 
has a mass of ~210 electron masses, may be either positive or negative, 
and is unstable, decaying into an electron and two neutral particles of 
negligible rest mass (presumably neutrinos) with a mean life of 2.15 
X 10 — 6 sec. However, subsequent investigation has shown that the 
muon has a very weak interaction with nucleons, a fact which eliminates 
it from the role of nuclear binding material. 

There is, however, another type of meson, the pi-meson or pion, which 
appears to be more directly associated with the nuclear forces. It has 
a mass of ~275 electron masses, and decays into a muon and (presum- 
ably) a neutrino, with a mean life of 2.6 X 10~ 8 sec. Pions have been 
produced in the interaction between two nucleons, or between a nucleon 
and a gamma-ray, when there is enough energy available to supply the 
rest mass. In addition to charged pions of both sign, there have also 
been observed neutral pions (whose mass appears to be slightly less 
than the mass of the charged pion) which decay into two gamma-rays, 
with a very short mean life (~10 — 15 sec). Finally, there is evidence 
for the existence of one or more types of still heavier meson. Very little 
is known concerning their interactions with nuclei. 

Of the various types of mesons which have been observed, it is the 
pion which most probably plays the role of nuclear binding agent. It is 
produced directly in the interaction between two nucleons, and between 
nucleons and gamma-rays; the cross section for its production is of an 
order of magnitude which indicates that it interacts strongly with nu- 
cleons — strongly enough to account for the strength of nuclear forces. 
Indeed, recent work on the production of pions, through the use of 
ultrahigh-energy machines, and on the interaction of pions with nucleons 
and nuclei provides the strongest arguments for the fundamental valid- 
ity of the meson theory of nuclear forces. 

Although meson theories have, so far, been unable to account in detail 
for the nuclear forces, they do provide a qualitative basis for under- 



240 The Neutron [Pt. VII 

standing the properties of nucleons (like, say, the anomalous magnetic 
moments of the neutron and proton) and of nuclear forces. The evi- 
dence on meson production at high energies together with the evidence 
on the charge independence of the nucleon-nucleon forces favor a 
symmetrical meson theory, in which charged mesons of both sign, as 
well as neutral mesons, play a comparable role. The exchange of charged 
mesons leads to the neutron-proton force, while neutral mesons are ex- 
changed in the neutron-neutron and proton-proton interactions. 

3. The Neutron-Electron Interaction. From the observation that the 
neutron has an intrinsic magnetic moment, it may be inferred that the 
neutron is a complex particle whose structure contains, at least for part 
of its existence, some moving charges. Indeed, according to current 
meson theories, the neutron can dissociate, spontaneously, into a 
tightly bound system of proton and negative pion, according to the 
reaction 

n +±p + 7T — 

During the fraction of its existence as a neutron proper, it can be re- 
garded as truly neutral; in the proton-pion state, however, the neutron 
has a charge structure corresponding to a relatively concentrated posi- 
tive charge surrounded, at a somewhat greater distance (of the order of 
a few times 10 — 13 cm), by a cloud of equal negative charge. The frac- 
tion of the time during which the neutron is in this dissociated state is 
variously estimated in the different meson theories to lie somewhere in 
the range ~0.1 to 0.5. 

As a consequence of this structure, there is an electromagnetic inter- 
action between neutrons and electrons, the predominant aspects of 
which can be described in terms of the interaction between two magnetic 
dipoles (that of the electron and that of the neutron). This magnetic 
dipole-dipole interaction is, of course, strongly dependent on the rela- 
tive orientations of the spins of the particles. It has been investigated 
extensively in studies of the scattering of slow neutrons by paramagnetic 
and ferromagnetic materials; these phenomena will be discussed in de- 
tail in Section 5. 

A magnetic interaction of the dipole-dipole type would, because of its 
spin-dependent character, vanish for the case of a neutron interacting 
simultaneously with an even number of electrons which are in a 1 »So 
state. Nevertheless, it has been observed that there is a small, spin- 
independent neutron-electron interaction. Such an interaction follows 
from meson-theoretic descriptions of the structure of the neutron, and 
its observed magnitude is in rough agreement with theoretical expecta- 
tions (to be discussed later in this section). The strength of the inter- 



Sec. 1C] Properties and Fundamental Interactions 241 

action is such that it leads to a (spin-independent) cross section for the 
scattering of neutrons by a single bound electron of ir,~5X 10~ 7 
barns. 

Although the value of <r e is exceedingly small as compared to the cross 
sections for scattering of neutrons by nuclei (~ a few barns), effects of 
this scattering are, nevertheless, detectable in the scattering of slow neu- 
trons by some heavy atoms. The possibility of observing such effects 
was first suggested and discussed by Condon (C24). This possibility 
results from the interference between the elastic scattering of sufficiently 
slow neutrons by the nucleus and by all the electrons in the atom. 

Let us consider the scattering of a beam of slow neutrons, wavelength 
of the same order as the atomic dimensions, by a single atom containing 
Z electrons. The differential scattering cross section is 

-£ = | a + Za e F(6) \ 2 (8) 

where «o an d a e are, respectively, the amplitudes for scattering of a neu- 
tron by the nucleus and by a single bound electron, 1 i.e., <r e = 4ira e 2 . 
F(d) is the atomic electron form factor, which can be computed from the 
electron distribution or, alternatively, obtained from data on the scat- 
tering of x-rays (C22). Since c — 4ira 2 5s> <r e , we may write 

For neutrons of wavelength large compared to the atomic dimensions, 
F(0) — > 1, while, as the neutron wavelength becomes small compared 
to the distance between the electrons, F(6) — > 0. To observe a possible 
neutron-electron interaction of the magnitude expected, it is absolutely 
necessary to take advantage of the interference between the neutron 
scattering by the nucleus and that by the atomic electrons. 

For an estimate of the magnitude of possible effects, let us assume a 
set of arbitrary, but reasonable, values, a ~ 5 barns, cr e ~ 5 X 10~ 7 
barns, ZF « 50. Then 

— = 2ZF ( — ) ~ 3 percent 
00 wo/ 

1 Strictly speaking, ao refers only to the coherent portion of the nuclear scattering 
amplitude. When there is, in addition, an incoherent contribution to the nuclear 
scattering, the above expression for the differential atomic scattering cross section 
requires the addition of an incoherent term, ai 2 , where the total nuclear scattering 
cross section is <r s = 4ir(ao 2 + ai 2 )- Although, for the discussion of this section, we 
shall assume that oi = 0, all the expressions to be derived can easily be modified 
if the nuclei exhibit any incoherent scattering. 



242 The Neutron [Pt. VII 

Although it should be possible to measure an effect of this magnitude, 
it is necessary to devise some means of distinguishing it from the purely 
nuclear scattering, whose cross section is seldom known to an accuracy 
of 3 percent. One possibility is the method devised by Fermi and Mar- 
shall (F23). They compared the scattering of thermal neutrons by 
gaseous xenon atoms, into a given solid angle, at angles of 45° and 135°. 
Such a difference would arise from two sources: (1) the difference in 
the electron form factor which was, for the neutrons employed, F(45°) 
— F(135°) = 0.261; (2) an asymmetry in the scattering due to the 
center-of-mass motion of the scattering system, since the angles were 
fixed in the laboratory system. In order to compute the magnitude of 
this effect, it is necessary to take into account the thermal motion of 
the xenon atoms (Doppler effect), as well as the energy distribution in 
the slow-neutron beam and the variation of the sensitivity of the de- 
tector with neutron energy. 

Fermi and Marshall observed a definite asymmetry in the scattering 
of ~2 percent which, however, became much less than the experimental 
uncertainty of 0.85 percent after they applied the Doppler effect cor- 
rection. Thus, although the accuracy of their measurement is insuffi- 
cient to confirm the existence of the neutron-electron scattering, their 
results set an upper limit to the neutron-electron interaction. If the 
interaction is described in terms of a fictitious square well potential of 
depth V and range equal to the classical electron radius, r = e 2 /mc 2 
= 2.8 X 10~ 13 cm, the experiment of Fermi and Marshall gives | V Q \ 
<, 5000 ev. (Compare with V <~ 25 X 10 6 ev for the neutron-proton 
potential.) l 

The method of Fermi and Marshall has been reapplied by Hamermesh, 
Ringo, and Wattenberg 2 (H28) with, however, significant improve- 

1 The spin-independent neutron-electron interaction is, if it is of mesonic origin, a 
short-range interaction. Consequently, the value of <r e for slow neutrons depends 
only on the volume integral of the interaction potential, rather than on any details 
of its shape. The Born approximation yields 

AW 

fl-fr 

where M is the neutron mass and 

f4w 



6 = 4x Cv(r)r 2 dr = - (~\ 



TW 



for the fictitious square well potential of range ro. A value of Vo = 5000 ev corre- 
sponds to a e = 4.3 X 10" 7 barns. It should be noted that a negative b (attractive 
potential) corresponds to a negative a,., while the nuclear scattering amplitude, Oo, 
is usually positive. 

2 I am indebted to Dr. A. Wattenberg for advance communication and discussion 
of these results. 



Sec. 1C] Properties and Fundamental Interactions 243 

ments in the geometry and a large increase in the number of neutrons 
counted. They have obtained the value 

V = 4100 ± 1000 ev 

This value is based on measurements of the scattering of krypton as 
well as of xenon; measurements were also performed on argon to check 
the correction for the center-of-gravity motion. Of the uncertainty of 
±1000 ev, quoted by these investigators, half is statistical and half is 
their estimate of possible systematic errors. Their result is in good 
agreement with the previous measurement of Havens, Rabi, and Rain- 
water (H46, H47). 

The method of observing the neutron-electron interaction, devised 
by Havens, Rabi, and Rainwater, is based on observation of the total 
cross section 



= fda (6) = <r ± 2ZF(cr cr e y A 



(9) 



They distinguish the effect of the coherent electron scattering by ob- 
serving the variation of a with neutron energy for wavelengths of the 
order of the atomic dimensions. For a neutron source of variable 
energy, they employed the Columbia velocity selector; as scattering 
materials, they used liquid lead and bismuth. Their method suffers 
from the difficulty that it is necessary to apply rather large corrections 
to the observed cross section vs. energy curve before the effect of the 
neutron-electron interaction can be ascertained. These corrections 
arise from a number of causes: (1) In addition to the nuclear scattering 
there is, for lead and bismuth, a small nuclear absorption whose cross 
section varies with the neutron velocity as 1/v. This correction can be 
applied with good accuracy. (2) Since they were dealing with atoms 
bound in a liquid, rather than with free atoms, it is necessary to take 
into account the effects of the binding (see page 231) on the variation of 
the cross section with energy. Although such effects (which include, of 
course, a Doppler effect for the motion of the atoms in the liquid) de- 
pend, to a certain extent, on the details of the binding mechanism, it has 
been shown by Placzek (P19) that they can be evaluated with the requi- 
site accuracy. (3) There are, in the scattering of neutrons by liquids, 
effects due to interference between the scattering by different atoms. 
Although such effects are relatively small, since the experiments involve 
neutrons with wavelengths of the order of atomic dimensions and hence 
somewhat smaller than the interatomic distances, they cannot be neg- 
lected in comparison with the effects of the neutron-electron scattering. 
The liquid coherence effects can be shown to vary as X 2 , in the energy 



244 The Neutron [Pt. VII 

region of interest, and the coefficient of the X 2 term can be evaluated by 
a general method for systems of high density and small compressibility, 
as shown by the calculations of Placzek, Nijboer, and van Hove (P21). 
(4) Although the three effects discussed above, all of which arise from 
nuclear causes, can be taken into account in a satisfactory fashion, there 
is a possible electronic effect which must be taken into consideration. 
Even though the ground states of atomic lead and bismith are 1 S , and 
Fermi and Marshall (F23) have proved that scattering due to the neu- 
tron magnetic moment is negligible for such atoms, small magnetic 
scattering effects might possibly be present when these atoms are in the 
liquid state. 

The results upon which Havens, Rabi, and Rainwater base their 
value of the neutron-electron interaction were obtained by careful 
analysis of data on the scattering of monoenergetic slow neutrons by 
liquid bismuth; they lead to 

V = 5300 ± 1000 ev 

Of the quoted uncertainty, 650 ev is statistical in origin, the rest being 
an estimate of possible systematic errors. 1 

Although a detailed meson-theoretic discussion of the spin-independ- 
ent neutron-electron interaction is outside the province of this review 
(even assuming that an adequate theory were available), the phenomena 
which can give rise to such an interaction are, however, susceptible of 
relatively simple physical interpretation. 2 Actually, there are two dis- 
tinct types of interaction which, together, can probably account for the 
observed effect. The first may be regarded as an electrostatic interac- 

1 D. J. Hughes has reported (at the February 1952 meeting of the American Phys- 
ical Society) a measurement of the neutron-electron interaction by observation of 
the critical angle for total reflection of cold neutrons from a liquid oxygen-bismuth 
interface (see Section 5). His preliminary results are in excellent argeement with the 
measurements of Havens, Rabi, and Rainwater and of Hamermesh, Ringo, and 
Wattenberg. However, this method appears to be capable of yielding the most 
accurate value of the strength of the neutron-electron interaction. Using this 
method, Harvey, Hughes, and Goldberg have obtained Vo = 4250 ± 400 ev (pri- 
vate communication, October 1952). 

2 Purcell and Ramsey (P32), in their discussion of the possible existence of a neu- 
tron electric dipole moment, point out that the observed neutron-electron interaction 
could, alternatively, be accounted for by the assumption of a small electric dipole 
moment of the neutron. The magnitude required to account for the observed inter- 
action is that of two opposite electron charges separated by a distance of 3 X 10 -18 
cm. Since the experiments of Purcell, Ramsey, and Smith (S40) have yielded an 
upper limit of two electron charges separated by 5 X 10 -21 cm for the electric dipole 
moment of the neutron, the explanation of the observed neutron-electron interaction 
lies, in all likelihood, along the meson-theoretic lines indicated below. 



Sec. 1C] Properties and Fundamental Interactions 245 

tion which arises from the fact that the neutron is not a strictly neutral 
particle but behaves, at least during an appreciable fraction of its 
existence, as a positive charge surrounded by a negative meson cloud. 
Thus, when the neutron and electron are separated by a distance less 
than the extent of the meson distribution (which is of the order of the 
classical electron radius r ) they will interact electrostatically. The 
predicted magnitude of this effect depends on the form of the meson 
theory used in its calculation and, in any event, has only been calcu- 
lated approximately (i.e., by perturbation techniques, to first order in 
the coupling constant). Such calculations have been carried out by 
Slotnick and Heitler (S39), by Case (C2), by Dancoff and Drell (D2), 
and by Borowitz and Kohn (B48). These calculations predict effects of 
the order of, but somewhat smaller than, the observed interaction. 
Thus, for instance, Case (C2) obtained an attractive interaction which, 
when expressed as a square well potential of range r , yields Vo = 
300(/ 2 /2t) ev; f 2 /2w, for this case (the coupling constant for a pseudo- 
scalar meson with pseudoscalar coupling), is ~5, giving V «* 1500 ev. 
The second effect arises, as was pointed out by Foldy (F37), as a 
direct consequence of the anomalous magnetic moment of the neu- 
tron. It can, in fact, be computed, without specification of the form 
of the meson theory, 1 by assuming that the neutron obeys the Dirac 
equation with the additional "Pauli terms" (P7) in the Hamiltonian, 
— n n (eh/2Mc)(JS(r -H — iPa-E). (n n ^ —1.91 is the neutron moment in 
nuclear magnetons.) The <r-H term leads to the spin-dependent, mag- 
netic dipole-dipole interaction between the neutron and the electron. The 
second term leads to an interaction of the form — yii„(eh 2 /4M 2 c 2 )|8 div E 
which corresponds, for the field of a point electron, to an attractive 
interaction with 

7o= -(T)(7) 2 (5) 2 " c2 = 4100ev (10) 

Thus, the Foldy term alone appears almost sufficient to account for the 
observed interaction. A more accurate measurement of the strength 
of the neutron-electron interaction should, indeed, eventually lead to 
an evaluation of the magnitude of the electrostatic interaction and 
permit a check on the applicability of specific meson theories. 

1 Since the fact that the neutron has an anomalous magnetic moment is, itself, a 
consequence of meson theories, it could not be said that this effect is of non-mesonic 
origin. Indeed, some of the meson-theoretic calculations (S39, D2) include the effect. 
However, since no existing theory is capable of yielding the observed value of the 
neutron magnetic moment, it appears more reasonable to follow the method of 
Foldy, which assumes the observed neutron moment, and to compute this effect by 
a phenomenological treatment. 



246 The Neutron [Pt. VII 

The two effects described above are susceptible of relatively simple, 
order-of-magnitude computations. The strength of the electrostatic 
neutron-electron interaction has been estimated by Fermi and Marshall 
(F23), who considered a model of a point proton (charge g 2 e) surrounded 
by a negative meson cloud of charge density 

' w --(^M-ir) <"' 



(13) 



This expression is suggested by Yukawa's original (scalar) theory of the 
meson field, according to which the extent of the meson wave function 
is essentially determined from the uncertainty principle, 

a ^ h/KC (12) 

where k is the pion mass («275 electron masses), g 2 ~ 0.1 to 0.5 is the 
is the fraction of time during which the neutron exists in the proton- 
meson state. 

The potential energy U(r) of interaction of the above charge distribu- 
tion with a point electron can be obtained by solution of the Poisson 
equation, and the strength of the neutron-electron interaction is given by 

b = 4x f U(r)r 2 dr = - (-) g 2 e 2 a 2 

Assuming a potential well of radius r and depth Vo, we obtain 
36 /g 2 \ /m\ 2 /hc\ 2 , 

Considering that the computation is classical and, furthermore, modeled 
on a meson theory which is known to be inadequate, the agreement with 
the meson-theoretic computations, quoted in the preceding, is quite 
satisfactory. 

It is somewhat more difficult to give a simple physical interpretation 
of the interaction which arises from the neutron's magnetic moment. 
However, Weisskopf l has suggested a rather ingenious derivation of 
the Foldy term: The assumption that the neutron obeys the Dirac equa- 
tion leads to a zitterbewegung in the motion of the neutron, such that its 
path is a spiral whose radius is of the order of the neutron's Compton 
wavelength, R ~ h/Mc. The neutron travels with the velocity of 
light, c, in this spiral path whose pitch is such as to give a transport 
velocity equal to the velocity v of the neutron. (The intrinsic spin 
angular momentum of the neutron can be interpreted as arising from 
this spiral motion.) Accordingly, when the neutron is within a distance 

1 Private communication. 



Sec. 2A1 Interaction with Nuclei 247 

R of the electron, which, we assume, behaves like a point charge, there 
is a magnetic spin orbit interaction between the electron current and 
the neutron's intrinsic magnetic moment whose energy is, neglecting 
numerical factors, 

'"-(s)^-^ (15) 

Since this interaction has a range of ~R, we obtain 

-ER 3 /hc\ 2 /m\ 2 , 

T --z-~-*k)Kz) m * (10s> 

which is, aside from the factor -f-, the result of the calculation of Foldy. 
Since the interaction depends only on the electron's charge, it is easily 
seen to be spin-independent. 



SECTION 2. INTERACTION WITH NUCLEI 

A. Introduction 

The experiments through which the existence of the neutron was 
established and in which its properties were first elucidated very soon 
indicated that neutrons are capable of inducing nuclear transmutations. 
Some of these early investigations have been discussed in Section 1. In 
these the neutron-induced transmutations were observed in cloud cham- 
bers. Although the cloud chamber is a very effective instrument for 
observing nuclear transmutations, its use imposes serious restrictions on 
the types of nuclear reactions which can possibly be detected: The tar- 
get nuclei must be capable of being introduced, as an appreciable con- 
stituent, into the cloud chamber; the transmutation products must be 
charged and have sufficient energy (range) to leave observable and 
identifiable tracks. With relatively few exceptions, cloud chamber 
techniques can only be applied to the study of nuclear transmutations 
induced in light elements by fast neutrons. 

With the announcement by Curie and Joliot (C41), in January 1934, 
of the production of artificially radioactive nuclei, an entirely new field 
of neutron investigation became available. 1 Their discovery prompted 

1 The discovery by Curie and Joliot was made in experiments involving the bom- 
bardment of boron, magnesium, and aluminum by polonium alpha-particles. They 
first reported the reaction 

AF + a. -> P 30 + n 

p30 _> gi 30 + e + 

and proved that the reaction product is an isotope of phosphorus, decaying by posi- 
tron emission, with a half-life of 3.25 min. 



248 The Neutron [Pt. VII 

Fermi to investigate the possibility of producing artificial radioactivity 
by neutrons, and he soon announced (F13) that the bombardment of 
aluminum and fluorine by neutrons does indeed induce such radioactiv- 
ity. Within a short time, Fermi and his co-workers at the University 
of Rome succeeded in inducing artificial radioactivity in a variety of 
elements through neutron bombardment (F12). In the next few years 
the Rome group played a leading role in the investigation of neutron- 
induced nuclear reactions. 

A significant step forward in the study of neutron reactions resulted 
from the observation by Fermi, Amaldi, Pontecorvo, Rasetti, and 
Segre (F14) that the neutron-induced radioactivity in silver can be sig- 
nificantly increased by interposing, between the source of fast neutrons 
and the silver detector, a slab of paraffin. This effect was correctly 
ascribed to the slowing down of the neutrons by collisions with hydrogen 
nuclei, and it indicated that the probability of neutron capture increases 
with decreasing neutron energy. This conjecture was confirmed by the 
observation, by Moon and Tillman (M35), of an effect of the tempera- 
ture of the moderator on the neutron capture. The slowing down of 
neutrons in hydrogen-containing materials will be discussed subse- 
quently. For the purpose of this summary it is, however, important to 
note that the recognition of the strong dependence of neutron capture 
probability on the neutron velocity was a significant step toward the 
understanding of neutron reactions and that, for many years, sources 
of neutrons slowed down in paraffin were the main tool for the investi- 
gation of neutron-induced nuclear reactions. 

The fact that radioactivity can be induced by neutrons in the heaviest 
elements is already a strong indication that the reactions responsible for 
their production probably do not involve the emission of charged parti- 
cles since, for heavy nuclei, the Gamow barrier is a very strong deterrent 
to charged particle emission. Furthermore, in many cases the radioac- 
tive nuclei were shown to be isotopes of the bombarded nuclei (F12). 
In particular, the ~15 hr half-life /3 — activity resulting from neutron 
capture by Na 23 , discovered by Bjerge and Westcott (B32), was identi- 
fied as an isotope of sodium by Amaldi, D Agostino, and Segre (A16) ; 
the fact that the decay involves emission of p~ particles was interpreted 
to imply that the radioactive isotope is Na 24 (rather than Na 22 ), indi- 
cating an (n,y) reaction. The observation that the probability of pro- 
ducing radioactivities in heavy nuclei increases with decreasing neutron 
velocity represents an even stronger argument for the contention that 
the reaction involved is that of radiative capture (n,-y) — capture of a 
neutron followed by gamma-ray emission. That this is indeed the case 
was shown by Amaldi, DAgostino, Fermi, Pontecorvo, Rasetti, and 



Sec. 2 A] Interaction with Nuclei 249 

Segre (A19), who proved that the capture of slow neutrons by cobalt, 
cadmium, chlorine, iridium, silver, and mercury is accompanied by 
gamma-ray emission. 1 For slow neutrons, radiative capture is usually 
the most important reaction. However, in the case of the capture of 
slow neutrons by light nuclei, charged particle emission may be much 
more probable. Thus, Chadwick and Goldhaber (C9) and, independ- 
ently, Amaldi, D'Agostino, Fermi, Pontecorvo, Rasetti, arid Segre 
(A17) showed that the strong capture of slow neutrons by lithium and 
boron is due to an (n,a) reaction. Somewhat later Burcham and Gold- 
haber (B75) demonstrated that the capture of slow neutrons by nitrogen 
is due to an (n,p) reaction. 

Perhaps the most important advance in the study of neutron reac- 
tions was the discovery of slow-neutron resonances — the preferential 
capture of neutrons of specific energies by certain nuclei. The first indi- 
cation of resonance effects was obtained in the experiments of Bjerge 
and Westcott (B33), soon followed by the work of Moon and Tillman 
(M35, T9), which established the resonance nature of some slow-neutron 
interactions. There then followed a period of rapid development of tech- 
niques for studying the properties of slow-neutron resonances. Among 
the most important of these were: the use of cadmium difference and 
self -indication methods by Fermi and Amaldi (F16) and by Szilard (S64) ; 
the measurement of neutron slowing down lengths in paraffin using, as a 
detector, the resonance in question, by Amaldi and Fermi (A20); the 
use of boron absorption techniques for the measurement of resonance 
energies by Frisch and Placzek (F45) and by Weeks, Livingston, and 
Bethe (Wll) ; the development of the first slow-neutron velocity selector 
by Dunning, Pegram, Fink, Mitchell, and Segre (D20). These tech- 
niques, their development and extension, will be discussed in detail in 
Section 3. Among the most significant of the early studies was the 
proof, by Dunning, Pegram, Fink, and Mitchell (D19), that the very 
large interaction of thermal neutrons with cadmium cannot be due to 
neutron scattering and must, therefore, be assumed to result from the 
radiative capture process. 

The study of slow-neutron resonance capture received great impetus 
from the theoretical considerations of Bohr (B44) on the role of the com- 
pound nucleus in nuclear reactions. On the basis of Bohr's arguments, 
it became possible to understand the existence of slow-neutron capture 
resonances, and to interpret their observed properties in terms of the 
Breit-Wigner formula (B60) (previously derived on the basis of very 

1 The emission of gamma-radiation in the capture of neutrons by hydrogen had 
previously been observed by Lea (L10). 



250 The Neutron [Pt. VII 

general considerations) for the energy dependence of the nuclear cross 
section in the neighborhood of a resonance. 

In addition to the work mentioned above, some of the investigations 
which were important in confirming the theoretical ideas concerning 
neutron resonances include those of Preiswerk and Halban (P30), 
Frisch (F46), and Goldsmith and Rasetti (G19). The experiments and 
theory of slow-neutron resonances, up to 1937, are summarized by Bethe 
(B24), by Bohr and Kalckar (B45), and by Moon (M36). The theory 
and observation of slow-neutron resonances, up to 1940, are discussed 
by Peierls (P10). 

While the most important advances in neutron physics were made in 
the study of the properties of slow neutrons, the investigation of fast 
neutron reactions was not completely neglected. Of the artificial radio- 
activities induced, by neutrons from Rn-Be sources, in light elements, the 
pioneering work of Fermi and co-workers (F12) showed, by chemical 
separation of the radioactive nuclei, that the responsible reactions were 
mainly (n,p) and (n,a). In the case of the heavy elements, the radio- 
active products appeared, in most cases, to be isotopes of the target 
nuclei. Fermi and co-workers advanced two alternative reaction possi- 
bilities: radiative capture, or the ejection of a neutron from the target 
nucleus by a neutron-neutron collision, i.e., the (n,2n) reaction. The 
dependence of the probability of most neutron reactions on the neutron 
velocity, discussed above, led to the assumption of the first hypothesis, 
confirmed by the observations on sodium (B32, A16) and other ele- 
ments (A19). However, in 1936, Heyn (H56) proved that, for neutrons 
of sufficiently high energy, the (w,2n) reaction could also be induced. 

In the meanwhile, Dunning and co-workers at Columbia University 
(D18, D19) were studying the reactions of fast neutrons with nuclei, 
using Ra-Be sources. In addition, with the discovery, by Szilard and 
Chalmers (S63), of the photodisintegration of beryllium by the gamma- 
rays from radium and its products, there became available neutron 
sources in the intermediate-energy range. These were exploited by a 
number of investigators (M22, L13, G25) for the study of neutron reac- 
tions in the 0.1 to 1 Mev range. Of primary importance in the study of 
fast-neutron reactions was the discovery, in 1933, of the possibility of 
producing strong fast-neutron sources by the use of particle accelera- 
tors through (d,n) reactions (Crane, Lauritsen, and Soltan, C30), and 
(p,n) reactions (Crane and Lauritsen, C31). Of special significance in 
this respect is the d + d reaction (Oliphant, Harteck, and Rutherford, 
()3), which can be used with relatively low-voltage accelerators. 

Among the early results of fast-neutron studies was the observation 
by Lea (Lll) of the excitation of gamma-rays by the passage of fast 



Sec. 2B] Interaction with Nuclei 251 

neutrons through matter. Lea showed that the production of gamma- 
rays increased with the atomic number of the material traversed, and 
that the observed effects were not due to a neutron capture process; 
he ascribed them to the excitation of nuclei in the inelastic scattering of 
fast neutrons, followed by nuclear de-excitation through gamma-ray 
emission. Similar results were obtained by Kikuchi, Aoki, and Husimi 
(K12). It was shown, by Danysz, Rotblat, Wertenstein, and Zyw 
(D3), by Ehrenberg (E2), and by Collie and Griffiths (C20), that fast 
neutrons are indeed slowed down, far in excess of what could be ac- 
counted for by elastic scattering, in their passage through heavy ele- 
ments. 

The last of the significant new neutron reactions to be uncovered 
was neutron fission, discovered by Hahn and Strassman (H4) early in 
1939. This discovery was the result of a long series of investigations 
which attempted to understand and interpret the results of the neutron 
bombardment of uranium, first reported by Fermi et al. (F12). Imme- 
diately after the announcement by Hahn and Strassman that isotopes 
of barium, lanthanum, and cerium are certainly among the products of 
the capture of slow neutrons by uranium, Meitner and Frisch (M23) 
showed that a fission reaction (division into two fragments of roughly 
equal mass) was to be expected from energetic considerations; such a 
reaction, they observed, is highly exoergic. Physical evidence for the 
expulsion of high-speed nuclear fragments in the slow-neutron fission of 
uranium was soon obtained by Frisch (F49) and by Joliot (Jll). The 
history of the discovery and early investigation of the fission reaction 
has been reviewed by Turner (T17). 

B. General Considerations 

Given a complete knowledge of the law of force between nucleons and 
of the law of motion (quantum mechanics) which governs their mutual 
interaction, it is conceivably possible to set up a program of computing 
the static properties of all nuclei and the detailed characteristics of the 
interactions between neutrons and nuclei. Actually, this program is 
still in its preliminary stages, if indeed the goal is at all attainable for 
any but the lightest nuclei. In the first place, the law or laws of force 
are not yet sufficiently well established. In the second place, their appli- 
cation to complex, heavy nuclei would involve the solution of the equa- 
tions of motion for the many-body problem, in which the components 
are closely spaced and strongly interacting (so that the approximation 
methods, which are so useful in problems involving the electrons in an 
atom, may not be applicable to nuclei). Finally, it is by no means cer- 
tain that specifically many-body forces (not detectable in or predictable 



252 The Neutron [Pt. VII 

from a study of the nucleon-nucleon interaction) do not come into play 
in the interaction of the nucleons in a nucleus. 

Nevertheless, a number of attempts have been made to understand 
the properties of nuclear (especially neutron) interactions in terms of 
approximations in which the nucleons in the nucleus are assumed to 
behave as more-or-less independent particles (A19, Bll, B21, Pll). 
The predictions of such models were soon found to be at strong variance 
with experimental observations, especially with regard to the properties 
of slow-neutron resonances; for, while these models predict strong ther- 
mal neutron capture cross sections and slow-neutron resonances, the 
widths of the levels and the spacing of levels, as predicted, were far in 
excess of those observed. Furthermore — and most significant — the 
independent particle models were unable to account for the predomi- 
nance of radiative capture over scattering in most slow-neutron reso- 
nances. 

To overcome these difficulties Breit and Wigner (B60) introduced the 
idea that the slow-neutron capture process must involve more than one 
of the nuclear constituents; their arguments were based mainly on an 
analogy with certain atomic and molecular phenomena. The conceptual 
and theoretical groundwork for the understanding of nuclear reactions 
was laid by Bohr (B44) in a work of classic and far-reaching significance. 
Bohr emphasized the necessity of going to the opposite extreme from the 
independent particle picture, and of recognizing that the nucleons in 
the nucleus interact very strongly with each other. Thus, a neutron, 
on entering the nucleus, very rapidly loses its identity as bombarding 
particle, sharing its energy among all the constituents of the nucleus. 
Accordingly, a nuclear reaction must be regarded as taking place in two 
distinct and separable stages. In the first, the incident particle is in- 
corporated into a compound nucleus, sharing its energy — kinetic and 
binding — with the rest of the nucleus. In the second stage, the com- 
pound nucleus gives up its excitation energy by any one of the possible 
means at its disposal. These means include radiation, particle emission, 
and neutron re-emission. In this competition among the various (ener- 
getically) possible de-excitation processes, radiation can compete very 
favorably with particle emission, especially in heavy nuclei, since parti- 
cle emission (including neutron re-emission) requires the concentration 
of a large fraction of the excitation energy into one particle, a process 
which is relatively improbable. Furthermore, because of the sharing of 
the excitation energy among the nuclear constituents, the compound 
nucleus has a relatively long lifetime, which results in comparatively 
sharp resonances. These ideas lend themselves to quantitative de- 
velopment in terms of a statistical theory of nuclear energy levels and 



Sec. 2B] Interaction with Nuclei 253 

of nuclear reactions (B45, B23, F43, W16, L2) ; they provide the basis 
for many theoretical investigations of the properties of slow-neutron 
resonances (B25, K2, S37, B61, W29, F27, F28, A6). In their extreme 
form, they suggest the liquid drop model of heavy nuclei, which has been 
so useful in understanding the dynamics of the fission process (B46). 

So successful were the ideas of Bohr, and their quantitative develop- 
ment, in understanding, correlating, and predicting the properties of 
nuclear reactions, as well as some of the general features of the stable 
nuclei (binding energies, curve of stability) that the independent parti- 
cle model was completely discredited. There remained, however, cer- 
tain features of nuclear behavior which, although they could be fitted 
into the general framework of the statistical model, required concessions 
in the direction of the independent particle model; thus, it was neces- 
sary to treat somewhat differently nuclei with odd and with even num- 
bers of nucleons. Furthermore, the success of the independent particle 
model, as applied by Schmidt (S5), in correlating nuclear magnetic 
moments indicated that the ground states, at least, of nuclei require 
such a treatment. 

The accumulation of nuclear data has recently led to the recognition 
by Mayer (M7) that many nuclear phenomena indicate the existence of 
a nuclear shell structure, first predicted by Elsasser (E4), thereby re- 
viving the independent particle model. The particular stability of 
nuclei containing 2, 8, 20, 28, 50, 82, or 126 neutrons or protons is borne 
out by a large number of nuclear data. A number of forms of the inde- 
pendent particle model have been invoked to derive the above "magic 
numbers" (F6, N2), of which the most successful has involved the 
assumption of a strong spin-orbit coupling, by Haxel, Jensen, and 
Suess (H49) and by Mayer (M8). Table 3 (after Mayer, M9) partially 
summarizes the predictions of this model. 

While the statistical model and the independent particle model repre- 
sent opposite extremes, it appears necessary nevertheless to recognize 
that both points of view are applicable, each in its domain, and that the 
two approaches are complementary. Thus, the independent particle 
picture appears to be required for the understanding of the ground and 
low-lying excited states of nuclei, whereas the statistical model is more 
appropriate for phenomena involving relatively large nuclear excitation, 
as is the case in almost all neutron reactions. Weisskopf (W17) has 
pointed out that the nuclear situation seems analogous to that of the 
electrons in a metal, where the electrons can be described in terms of an 
independent particle model for the states of small excitation because of 
the effects of the Pauli principle, despite the fact that they are inter- 
acting very strongly with each other. 



254 



The Neutron 
TABLE 3 



[Pt. VII 



Order of Energy Levels Obtained from Those of a Square Well 
Potential by Spin-Orbit Coupling 







(After 


Mayer, M9; 






Oscillator 


Square 
Well 
Level 


Spin 


Number 


Number 


Total 


Number 


Term 


of States 


in Shell 


Number 





Is 


lSi A 


2 


2 


2 


1 


lp 


ipy* 


4 










lPH 


2 


6 


8 


2 


Id 


ldtft 


6 




(14) 




2s 


&X 


2 










ld% 


4 


12 


20 


3 


1/ 


V» 


8 


8 


28 




2p 


2PK 


4 
6 










2PK 


2 


(12) 


(40) 


4 


1(7 


ig% 


10 


22 


50 




2d 


2d H 

^g% 

2d H 
3sy 2 


6 
8 
4 
2 






5 


Ih 


lhi$i 


12 


32 


82 




2/ 


2/54 


8 








3p 


3pk 
3ph 


4 

6 

2 

10 






6 


3d 
4s 


2?« 


14 


44 


126 



Sec. 2B] Interaction with Nuclei 255 

Nevertheless, it is necessary to observe that neither model provides a 
complete description of nuclear phenomena in any energy region; the 
strong interaction undoubtedly influences the properties of the low- 
lying nuclear levels, while (as we shall have many occasions to observe) 
the shell structure has important effects on the characteristics of neutron 
reactions, even at relatively high levels of excitation. 

The properties of nuclear reactions have been discussed in Part VI. 
In this section we shall review these properties as they apply to neutron 
reactions, and summarize the status of the present knowledge of the 
interactions of neutrons with nuclei. The available data on neutron 
cross sections have been compiled by Goldsmith, Ibser, and Feld (G20) 
(October 1947), by Adair (A2) (July 1950), and by an AEC Committee 
(A0) (1952). The discussion here will lean heavily on these compila- 
tions, in which references are given to the original investigations. 

1. Energetics of Neutron Reactions. When a neutron of kinetic 
energy E is captured by the target nucleus, the product (compound) 
nucleus acquires an excitation of energy E' = E + e. e is the binding 
energy of the neutron in the product nucleus. This excitation makes 
possible a variety of nuclear reactions, depending on the particle or par- 
ticles whose emission from the compound nucleus requires less than the 
energy E'. The characteristics of the reactions observed depend on the 
properties of the compound nucleus at the excitation energy E', and on 
the competition between the possible modes of de-excitation of the 
compound nucleus. 

Clearly, for neutrons of kinetic energy E <SC e, the characteristics of 
the observed reactions will be critically dependent on the value of e. 
The variation of e with Z and A may be summarized as follows : For 
nuclei of A < 20, e exhibits large (periodic) fluctuations from nucleus 
to nucleus. For nuclei of A "> 20, the values of e, on the average, in- 
crease slowly from -~8 Mev to ~8.5 Mev at A — 130, and then decrease 
slowly to <~7.5 Mev for the heaviest nuclei. However, these values 
apply to the stable nuclei; the capture of a neutron by a stable nucleus 
usually results in a radioactive product nucleus for which the neutron 
binding energies are somewhat smaller than for the stable nuclei. Thus, 
the values of e, for captured neutrons, average ~7-8 Mev for A be- 
tween 20 and 150, falling slowly to ~6 Mev for neutrons captured by 
the heaviest nuclei. 

However, neutron binding energies fluctuate significantly from nucleus 
to nucleus. The capture of a neutron by a nucleus with an odd number 
of neutrons results in a greater (by ~l-2 Mev) release of energy than 
capture by adjacent nuclei of even neutron number. Quite large devia- 
tions from the average may also occur in the vicinity of neutron magic 



256 The Neutron [Pt. VII 

numbers; thus, the value of e for a neutron captured in a nucleus lack- 
ing but one neutron for a closed shell will be anomalously high, while « 
for a neutron captured by a nucleus already having a completed neutron 
shell is anomalously low. Furthermore, such magic number effects 
appear to persist for neutron numbers ranging considerably to either 
side of a magic number (K23). 

In those (relatively rare) cases in which the masses of the target and 
product nuclei are known, the neutron binding energy can be computed : 

e = 931.16 {M(X A ) + M(n) - M(X A+1 )} in Mev (16) 

For light nuclei the masses of the nuclei involved are, in general, avail" 
able and the above computation can be made. For nuclei of A "> 30 
the masses are usually not known, and it is then necessary to resort to 
the general considerations described above, if it is desired to estimate 
the value of «. 

From the above expression for e, it may be seen that it is not necessary 
to know the absolute value of the nuclear masses in order to compute e, 
but only the mass difference, M(X A+1 ) — M(X A ). Nuclear mass dif- 
ferences are frequently known as a result of measurements of the energies 
of reactions involving the two nuclei in question, even when the absolute 
values of the masses are unknown. Thus, in the region of the naturally 
radioactive nuclei, neutron binding energies can be computed with the 
aid of the observed energies of alpha- and beta-emission (W9). 

For a few nuclei the energy release accompanying slow-neutron cap- 
ture has been measured directly by observation of the energy of the 
gamma-radiation emitted when the excited compound nucleus decays to 
its ground state. In this fashion, Bell and Elliott measured the binding 
energy of the deuteron (B16). Unfortunately, for nuclei more compli- 
cated than the deuteron the radiative de-excitation of the compound 
nucleus is usually achieved by the emission, in cascade, of a number of 
gamma-rays. Since there are usually a large number of energy levels 
of the compound nucleus available to such cascade gamma-ray emis- 
sion, the capture gamma-ray spectrum is quite complicated. However, 
in certain favorable cases the emission of a single gamma-ray, carrying 
away all the excitation energy, occurs in a reasonably large fraction of 
the decays; the energy of this gamma-ray gives, directly, the value of e. 
Such direct transitions to the nuclear ground states have been observed 
in a number of nuclei by Kinsey, Bartholomew, and Walker (K15). 

In general, the nuclear reactions in which we are interested can be 
represented symbolically as 

on 1 + Z X A -> ( Z X A+1 )* -* z x a + z _ z Y A+l - a + Q (17) 



Sec. 2B] Interaction with Nuclei 257 

[For many light nucleus reactions the intermediate (compound) nucleus 
step should be omitted, since it has no independent existence.] The re- 
action is frequently abbreviated as X. A (n,x)Y A+1 ~ a , or simply as 
~K. A (n,x). The Q value of the reaction is determined by the masses of 
the nuclei involved : 

Q = 931.16{AT(n) + M(X) - M(x) - M(Y)} in Mev (18) 

or, conversely, a measurement of the Q value can be used to determine 
one of the masses, if the other three are known, or the mass difference 
M (X) — M (Y), from the masses of the neutron and the ejected particle. 
A positive Q value indicates an exoergic reaction, i.e., a reaction that 
can take place (from the point of view of available energy) with neutrons 
of zero kinetic energy; a negative Q value means that the reaction is 
endoergic and is not possible for neutrons of kinetic energy less than a 
certain threshold value: 

f[M(X) + M(n)]) (04 + 1)1 

The factor in brackets arises from the conservation of momentum, which 
requires that some of the neutron kinetic energy be expended in pro- 
viding for the center-of-mass motion of the system. The possibility 
that a given reaction can be initiated by thermal neutrons depends on 
having Q > 0. Two reactions are, however, always possible for thermal 
neutrons: (1) elastic scattering (or re-emission) of a neutron (n,n), for 
which Q = 0; (2) radiative capture (n,y), for which Q = e. 

2. Theory of Neutron Reactions ; the Compound Nucleus. According 
to the Bohr model, most neutron reactions can be regarded as taking 
place in two distinct stages — the formation of an excited compound nu- 
cleus, and its subsequent decay. In our survey of the theoretical conse- 
quences of these ideas, we lean heavily on the schematic theory of nuclear 
reactions developed by Weisskopf and co-workers (F27, F28, W17). 
This theory is based on some general assumptions regarding the struc- 
ture of nuclei: that the nucleus has a well-defined surface, which is a 
sphere of radius R, outside of which the interaction with neutrons is 
negligible; that, once inside the nucleus, the neutron interacts very 
strongly with the nuclear constituents, rapidly sharing its excitation 
energy among them. 

Because of the saturation property of nuclear forces, the average 
distance between nuclear constituents is independent of their number, 



258 The Neutron [Pt. VII 

A. Thus, the volume V of a nucleus is proportional to A, and its radius 
can be approximated by the relationship 

R = r A 1A (20) 

with r between 1.3 and 1.5 X 10~ 13 cm. 

A neutron of kinetic energy E will, on penetrating the nuclear sur- 
face, suddenly find itself in a region where its kinetic energy is E' ^ E 
+ e, where e is the average kinetic energy of a nucleon inside the nu- 
cleus. Such a sudden change of kinetic energy results, from wave- 
mechanical considerations, in a reflection; the probability for penetra- 
tion of the surface is given by 

4JcK 
P = -tt-t^-, (2D 



where 



(k + KY 

„ 2ME 1 

k 2 = — — = - (22a) 



X 



2 



for the incident neutron, and 

K2 = ™<l+A = k2 + K j ,., w. 

fi 2 h 2 

for the neutron inside the nucleus. (K and k are the wave numbers 
associated with the neutron inside and outside the nucleus, respectively.) 
The value of K can be computed on the assumption that the nucleons 
in the nucleus obey Fermi-Dirac statistics, distributing themselves 
among the various possible states of momentum, spin, and isotopic spin 
(charge) so that no state is occupied by more than one nucleon, giving 

£ = ( ) ( — ) ^ 20 Mev 

V 2M I \16ttVJ 

(23) 
K ^ 1.0 X 10 13 cm -1 

for r = 1.5 X 10 -13 cm. For slow neutrons, k « K ^ K , and 

4fc 
P ^ — « 1 (24) 

Ko 

Despite the smallness of P for slow neutrons, the probability of neutron 
capture to form a compound nucleus may be quite large since slow 
neutrons have a large extent (wavelength). The capture probability 
depends critically on the proximity of the excitation energy (E' = 
E + e) to one of the energy levels of the compound nucleus, being 
essentially a resonance phenomenon. We postpone the discussion of 



Sec. 2B] Interaction with Nuclei 259 

the formation of the compound nucleus to the next section and confine 
ourselves, at this point, to consideration of what happens after the com- 
pound nucleus has been formed. 

Once a neutron has entered the target nucleus, it very rapidly shares 
its energy among the nuclear constituents, forming a compound nucleus 
in an excited state. Furthermore, owing to the smallness of P (which 
works both ways), the neutron has a very small probability of re-emerg- 
ing, even if it should find itself in the relatively improbable circumstance 
of possessing all the excitation energy. Consequently, the lifetime of 
the compound nucleus is very long compared to the time required for 
its traversal by a nucleon. In fact, for slow neutrons captured in nuclei 
containing a relatively large number of constituents (say A >^ 100) the 
most favorable mechanism for de-excitation of the compound nucleus is 
usually the emission of gamma-radiation, which is itself a relatively 
improbable process. 

These considerations can be made somewhat more quantitative. The 
compound nucleus, formed in an excited state, may have a number of 
(energetically) possible modes of decay which always include radiation 
(n,y), neutron re-emission (n,ri), and may also include the emission of 
one or more particles (n,a), (n,b), etc. These compete freely with each 
other, each mode of decay being characterized by a mean-life, t,- (the 
lifetime which the excited state would have if all other possible modes of 
decay were turned off). The mean-life of the excited state 

1 ^ 1 

-=£- (25) 

T ,■ Ti 

is associated, according to the uncertainty principle, with a finite width 
(energy spread) of the excited state of the compound nucleus: 

h 

r = - (26) 

T 

Correspondingly, we can define a partial width for the z'th mode of 
decay: 

h 
r,- = - (27) 

so that 

r = Er f (28) 

i 

The characteristics of the compound nucleus decay are determined by 
the relative values of the partial widths for the various possible modes 
of decay. For the case of gamma-ray emission, the entire excitation 



260 The Neutron [Pt. VII 

energy is available to the gamma-radiation. Since, for slow neutrons, 
e » E and E' « e , the probability of gamma-ray emission is essentially 
independent of the energy of the incident neutron. However, T y « 
constant may be expected to depend to a certain extent on the angular 
momentum properties of the state involved, so that the constant will 
not be the same for all the states of the compound nucleus. 

For particle emission, however, the situation is quite different. In 
this case, in addition to kinetic energy, the emerging particle must also 
be supplied with its binding energy. Since the excitation energy is 
shared among the many particles in the compound nucleus, the possi- 
bility of emission of a particle depends on the concentration of sufficient 
energy into this particle to allow it to escape. Let T a be the average 
time between such rearrangements of the nuclear constituents as would 
permit the emission of particle a. The frequency of emission of the 
particle a (l/r a ) is then given by the product of the frequency of such 
favorable configurations (1/T a ) and the probability that the particle a, 
given the requisite amount of energy, can penetrate through the nuclear 
surface : 

1 Pa 

- = V (29) 

Ta i-a 

Hence 

h hP„ 

r a = - = — (30) 

T a ±a 

According to the previous considerations, and- assuming that the kinetic 
energy of the emerging particle is small compared to its energy in the 
nucleus, and that it has zero angular momentum, 

4fc„ 
P a = — Qa (31) 

-fro 

where G a is the Gamow barrier for particle a; G n — 1 for neutrons. 
Thus 

4hG 
r„ ^ — — K = 7a G a E a * (32) 

The above relationship applies only to the emission of particles of 
zero angular momentum. It involves the kinetic energy of the emerg- 
ing particle, E a , through the factor E a y * and through the energy de- 
pendence of the Gamow factor, (?„, when the particle is charged. It 
involves the properties of the compound nucleus through the factor K 0) 
which is essentially independent of the nucleus involved, and the perio- 
dicity T a . The last factor depends quite critically on the specific nu- 



Sec. 2B] Interaction with Nuclei 261 

cleus involved and on the excitation energy of the compound nucleus 
state; this dependence on the properties of the compound nucleus is 
through the nuclear level spacing, at the excitation energy involved, as 
indicated by the following simplified considerations, due to Weisskopf 
(W17): Assume that the energy levels of the compound nucleus are 
equally spaced, 

E n = E + nD (33) 

Owing to the strong interaction, the nuclear wave function is given by 
a linear combination: 

£, -iE n t -iE t * -inDt 

* = 2-, a n<l>n exp — - — = exp — - — 2-, a n<t>n exp — - — (34) 

n=0 U B- n=0 h 

The periodicity T, which is defined as the time required for the nucleus 
to return to a previous configuration, is given by the condition 

| *(t + 2xh/D) | = | ¥(t + T) | = | ¥(*) | (35) 

whence 

2irh 
T = — (36) 

Although the level spacing in the compound nucleus is far from uni- 
form, the above relationship nevertheless holds approximately for our 
case, where D a is the average level spacing (of the levels which are 
capable of decaying through emission of the particle a in a state of 
given angular momentum) in the energy region of the excitation energy. 
Combining Eqs. (32) and (36), 

2G a D a 

r« « — k a = CD a G a E a 1A (37) 

for the emission of particles of zero angular momentum. The above is 
a much simplified derivation of a well-known relationship between level 
spacing and level width (B21, B46, F27), which is borne out by the 
observed neutron widths for slow neutrons (W31, B34). 

The E y% dependence of the particle width on its kinetic energy also 
follows from very general considerations involving the density of mo- 
mentum states available to the outgoing particle, in the case of emis- 
sion of zero angular momentum particles. For the emission of particles 
of higher angular momentum, I, there is an additional energy depend- 
ence, which can be expressed as follows: 

r„; = (21 + l)CD al G a iE a ^v al (38) 



262 The Neutron [Pt. VII 

where v a i = vi(k a R), R being the nuclear radius. Expressions for vi{x) 
are (F27, B34) 

f o = 1 



(39) 





x 2 






v l 


(1 + x 2 ) 

X 4 




V% 


(9 + 3x 2 


+ 


* 4 ) 



v 3 = 



(225 + 45z 2 + 6x 4 + a; 6 ) 
These have the limiting values 



y2l 



vi — > 1 for x — > oo 

Thus, for the emission of low-energy particles (k a R « 1), we need con- 
cern ourselves only with I = 0. At sufficiently higher energies, states 
of i! > come into play. 

Because of the selection rules associated with angular momentum 
changes, the absorption and emission of particles of different angular 
momenta usually involve different energy levels of the compound nu- 
cleus. For a given energy level of the compound nucleus, the emission 
of a given particle in different states of angular momentum can usually 
be treated on the same basis as the competition for the emission of dif- 
ferent particles. 

So far, the spacing of the levels in the compound nucleus, D, has en- 
tered our considerations only as a parameter in the level width formulas. 
Clearly, the level spacing is also of primary importance in the formation 
of the compound nucleus, since the probability of neutron absorption 
depends on the proximity of the nuclear excitation energy to one or 
more of the levels of the compound nucleus. 

On the basis of the Bohr model, it is possible to obtain a rough, semi- 
quantitative picture of the dependence of the level spacing of the nu- 
cleus on the excitation energy. For low excitation energies (of a few 
Mev or less) the possible modes of excitation of the compound nucleus 
are few, and the energy levels are widely spaced. The excitation of the 
low-lying levels will involve only one or a few nucleons, and the details 
of the low-lying levels will depend on the nucleus involved, being 



Sec. 2B] Interaction with Nuclei 263 

strongly influenced by such phenomena as magic numbers. As the en- 
ergy of excitation is increased, it becomes possible to excite more modes, 
involving a greater number of nuclear constituents, and the levels be- 
come more closely spaced. For large excitation energies (say ~10 Mev) 
an expression for the spacing of energy levels may be obtained from 
thermodynamic considerations, and is of the form 

w(E') = Cexp2(aE') 1A (41) 

w(E') = 1/D is the level density (number of levels per unit energy inter- 
val) at the excitation energy E'. Typical values of C and a, for a number 
of odd values of A, are given in Table 4; the level densities for even A 
nuclei appear to be somewhat lower. 

TABLE 4 

Constants Determining the Level Spacings of Some Odd A Nuclei, 

Eq. (41) 

(Derived by Blatt and Weisskopf, B34, from available data) 

A C (Mev" 1 ) a (Mev" 1 ) 



27 


0.5 


0.45 


55 


0.33 


1.3 


115 


0.02 


6.5 


201 


0.005 


12 



Figure 4a is a schematic energy level diagram for two nuclei, A ~ 55 
and A = 201. Aside from the details, which depend on the nucleus in- 
volved, the general characteristics of the level scheme for the heavier 
nucleus can be derived from that of the lighter by a contraction of the 
energy scale. This feature is qualitatively expected from the Bohr 
model since the heavier the nucleus, the more particles are available to 
participate in the excitation, and the greater the level density at a given 
excitation energy. Figure 4b shows the same level diagrams greatly 
magnified in the region of excitation corresponding to the absorption of 
a neutron. 

As implied by Fig. 4b, the level spacing of interest, in considering 
reactions involving neutron capture, is that which corresponds to the 
excitation E' — E + « of the compound nucleus. At first glance it 
might be expected that, for neutrons of moderate energy, this would be 
primarily determined by the binding energy, e. However, evidence has 
recently been presented by Harris, Muehlhause, and Thomas (H40) and 
by Hurwitz and Bethe (H81) that this is not the case. Thus, Harris 
et al. have presented data which indicate that the level densities at E' 



264 



The Neutron 



[Pt. VII 



10 



-P 5 







AZ55 














































































e~ 


















































A = 201 







































































































































































































































































Fig. 4a. Schematic energy level diagrams for two nuclei, A =* 55 and A « 201. 
The average level spacings are according to the statistical theory (Eq. 41), and the 
constants are from Table 4. The details of the structure, i.e., the relative positions 
of adjacent levels and the fluctuations in level spacing, are entirely fictitious, cor- 
responding to no known nucleus. They are intended to illustrate the possibility 
of large deviations from the average. 



Sec. 2B] 



Interaction with Nuclei 



265 



« e, for capture of neutrons by odd proton-even neutron and by even 
proton-odd neutron nuclei, are essentially the same (for nuclei of roughly 
the same A) although e is considerably smaller for the former than for 



Mev 
1.0 



0.9 



0.8 



0.7 



0.6 



6q 

II 
&1 



0.5 



0.4 



0.3 



0.2 



0.1 







A ==55 



AS 201 





















































































































































Magnification 






















X 10 




X 20,000 



























































































































































































































500 



400 



300 



200 



100 







Fig. 4b. Energy level diagrams of Fig. 4a greatly magnified in the region of the 

neutron binding energy. 

the latter (by ~1 Mev). They ascribe the difference in e to a lowering 
of the ground-state energy of the compound nucleus of even Z-even N, 
as compared to that of odd Z-odd N; the level densities corresponding to 
e are, however, the same for both types. Hurwitz and Bethe concur and 



266 The Neutron [Pt. VII 

present arguments for the hypothesis that the level density of the com- 
pound nucleus, at E' « e, is determined primarily by the binding energy 
of the target nucleus (rather than by that of the compound nucleus); 
target nuclei with high binding energy (e.g., even-even or magic number 
nuclei) result in compound nuclei with low level spacing. Data of 
Hughes and co-workers * on the radiative capture of fast neutrons, how- 
ever, appear to be in disagreement with this point of view. 

In any event, it should be pointed out that the effect of the apparent 
validity of the independent particle model at low excitation energies will 
be to invalidate the applicability of Eq. (41) to excitation energies of a 
few Mev. Available evidence, mainly from inelastic scattering and (p,n) 
reactions, indicates that level densities at low excitation energies are 
rather smaller than predicted by Eq. (41), and that the exponential be- 
havior does not really set in until values of E' >, 3-5 Mev are achieved. 
The constants of Table 4 should be regarded as applying, roughly, to 
energies E = E' — a ~ 0-2 Mev. 

The compound nucleus theory of nuclear reactions, outlined above, 
will be expected to break down at very high excitation, for at high 
enough energies the probability of escape of the incident neutron after 
a single traversal may be quite large, i.e., the nucleus is more or less 
transparent to high-energy neutrons. Thus, the incident neutron may 
not have sufficient opportunity to share its energy among the nuclear 
constituents before it (or one of the other nucleons) escapes. In this 
respect, it is to be expected that the absorption of a charged particle by 
a heavy nucleus will, even for high incident energies, be more likely to 
lead to a true compound nucleus, since the escape of the charged particle 
from the nucleus is impeded by the Gamow barrier. 

3. Formation of the Compound Nucleus; Resonances and the Breit- 
Wigner Formula. In accordance with the two-stage picture of neutron 
reactions (formation of a compound nucleus; subsequent emission of a), 
the (n,a) cross section is 

<j{n,a) = <r c — (42) 

The first factor, <r c , is the cross section for the absorption of the incident 
neutron into an excited compound nucleus; the second factor represents 
the relative probability for the de-excitation of the compound nucleus 
through the emission of a. 

The cross section for the formation of the compound nucleus depends 
critically on the proximity of the excitation energy (E + e) to one or 

1 Private communication. 



Sec. 2B] Interaction with Nuclei 267 

more of the energy levels of the compound nucleus. In the case where 
only one level of the compound nucleus is involved (all other levels 
being far enough removed from the value of the excitation energy so 
that their influence on a c may be neglected), the energy dependence of 
<r c is given by a characteristic resonance formula, 



r„r 

<r c = ttK*1 — ■ 

(E - i 

whence 



<r c = tX 2 / ~-^r — (43) 

J (E - E T f + r 2 /4 



a{n,a) = irX 2 / ~ — (44) 

K (E - E r ) 2 + T 2 /4 

Equation (44) is the well-known Breit-Wigner one-level formula; the 
symbols have been previously defined, except for E T , which is the neu- 
tron kinetic energy for which the excitation energy is equal to the dif- 
ference between the ground state and the energy of the compound nu- 
cleus level involved in the reaction, and /, a statistical weighting fac- 
tor which depends on the neutron spin, s = \ ; its orbital angular mo- 
mentum, lh; the total neutron angular momentum quantum number 
j = I ± \ ; the spin of the target nucleus, J; and the spin of the com- 
pound nucleus level involved, J = I+j,I + j— 1, • • -, \ I — j\: 

(2/+1) _p/ + l) 
(27+l)(2s+l) 2(27+1) 

The factor / is simply a measure of the probability that the neutron angu- 
lar momentum, jh, and that of the target nucleus, lh, either or both 
assumed oriented at random (unpolarized), will have a relative orienta- 
tion appropriate to the necessary total angular momentum, Jh, required 
for the formation of the compound nucleus level involved in the reso- 
nance. For slow neutrons, I = 0, j = \, and J = I ±\; whence for 
I ± 0, 



1 



1 

1 ± 



1 



f^/ . 



for large I (45a) 



2L (27+l)J 2 

and /o = 1 for 7 = 

For neutrons of I > there are two possible "channel" spins, j = I 
± |, each of which combines with the target nucleus spin 7 to give the 
possible compound nucleus spin values J. Some of these can be formed 
through only one channel; for these, the statistical factor, fi(J), is un- 
ambiguously given by Eq. (45). Other J values can be formed through 
both channels; for such levels the question arises as to whether the 
proper value of the statistical factor is fo(J) or 2f t (J). 



268 The Neutron [Pt. VII 

Let us consider a specific example: Assuming a P-resonance (1=1) 
for a target nucleus of / = §, the possible J values of a compound nu- 
cleus state are 

J = 2,1 fori = f 

J = 1,0 forj=f 
The statistical weights are, from Eq. (45), 

/i(2) = f 
/i(l) = f 
/i(0) = i 

For levels of J = 2 or J = 0, there is no ambiguity. The J = 1 state 
illustrates the possible ambiguity. On the one hand, a given J = 1 
level may be such that it interacts only with one of the two possible 
channel spins (or one of two possible orthogonal linear combinations of 
/ = -J and j = |); under these circumstances, the statistical factor 
/i(l) = i- O n the other hand, the interaction might be completely 
independent of j, depending only on the I value, which is the same for 
both channels. In this case, /i(l) = 2 X f = f • 

In either event, the ambiguity disappears if many levels of the com- 
pound nucleus are involved in a given interaction, since the sum of the 
statistical factors, over all possible J and j values, is ^2fi(J) = 3. This 
is a special instance of the general relationship J,i 

T,fi(J) = (2l+l) (45b) 

The situation with regard to the statistical factor is even more com- 
plicated if the same compound nucleus state can be formed by different 
neutron orbital angular momenta (12). However, this added compli- 
cation is seldom of practical importance, since the neutron widths will 
be very different for different values of I, which fact usually effectively 
eliminates the influence of all but the lowest possible I value. In addi- 
tion, for a given compound nucleus state the parities of both the initial 
and the final states are well defined. Hence, the choice of possible I 
values is even further limited, adjacent values not being simultaneously 
available. 

According to Eqs. (43) and (44), neutron capture and the accompany- 
ing reactions are characterized by sharp resonances for incident neutron 



Sec. 2B] Interaction with Nuclei 


269 


energies E = E r , with the maximum (peak) cross section 




47rX 2 / z r ni 
cole - r 


(46) 


4irX 2 fiT n iT a 
<r i(n,a) = 


(47) 


and (full) width at half maximum 




h 

ae = r = - 

T 


(48) 



Equation (44) applies to the case of elastic scattering as well as to the 
case of emission of some other particle. However, in addition to the 
scattering which involves the formation of a compound nucleus (so- 
called capture scattering) there is a second type of elastic scattering 
process known as potential scattering. Potential scattering results from 
the small penetrability of the nucleus to slow neutrons (Eq. 24) and can 
usually be described in terms of the scattering of an impenetrable sphere, 
for which the scattering cross section would be 

op = ttX 2 £ (21 + 1)| e** - 1 | 2 = 4*-X 2 £ (21 + 1) sin 2 & (49) 

i i 

|j is the phase shift of the scattered partial wave of orbital angular 
momentum Zh. For slow incident neutrons, R <5C X = 1/fc and 

So = kR & >0 « So (50) 

whence 

<r p ^ a p0 = 4xX 2 sin 2 kR ^ 4irR 2 (49a) 

The phase shifts corresponding to scattering by a rigid sphere can be 
obtained from the following general expression (F27, L23), with kR = x 

where J and N are, respectively, the Bessel and Neumann functions. 
So = x 



x _ 

Si = x h cot x 



_. x 2 - 3 

£ 2 = x — x + cot ■ 

Sx 

3x x(x 2 - 15) 

£q = a: H cot — 

?3 2 fa 8 - 15 



(50b) 



270 The Neutron [Pt. VII 

For x « 1, 

So = x 



x 3 x 5 x 7 






/x b x 1 \ 
\7 21/ 

) 



(50c) 



* 3 ~225\7 ~ 7' 



A more useful approximation is, in the limit x <JC | 21 — 1 |, 



x 2l+l 



& « tan -1 (50d) 

At the other extreme, for x 5S> I 2 , 

It 
6 « a: - - (60e) 

Values of the phase shifts vs. x have been tabulated by Lowan, Morse, 
Feshbach, and Lax (L23). 

Equation (49) applies to the general case of scattering by an arbitrary 
potential (as well as to scattering by a rigid sphere) : 

<r,c = 4ttX 2 J2 W + 1) sm 2 ft (49') 

i 

The phase shifts, ft, are obtained from the asymptotic behavior of the 
radial functions, u h in terms of which the wave function of the system is 
written : 

ii>(r,0) = 22 Ui(r)Pi(cos 0) (49") 

i 

The functions ui satisfy the radial Schrodinger equations: 



«i" + 



;?>— ^ 



ui = (49"a) 



For slow and intermediate neutrons, X ^> R and only I = scattering 
need be considered; i.e., ft> <3C /3 , and c sc = 4xX 2 sin 2 /3 . 

The value of ft, depends, of course, on the form of the scattering po- 
tential V (in Eq. 49"a, with 1 = 0). A number of possibilities are illus- 
trated in Fig. 5 in which u is plotted in a number of hypothetical cases. 



Sec. 2B] Interaction with Nuclei 

u (r) 



271 




Fig. 5. Illustrative examples of the joining of the neutron wave functions at the 
nuclear boundary, leading to different phase shifts and scattering lengths. Case 
(a) applies far from a resonance, on the assumption of a "rigid sphere" nuclear 
model. Cases (b), (c), and (d) represent resonance scattering. Case (e) is intended 
to illustrate scattering by a finite potential well. 



272 The Neutron [Pt. VII 

It is assumed that the potential V acts only in the range — R; foj 
r > R, u (r > R) = sin (kr + p ). The intercept a of u (r > R) on th< 
r axis, indicated with an arrow in each case, determines /3 = — a/%. 
a is called the scattering length or scattering amplitude; according tc 
convention, positive a will be assumed to mean that the intercept is on 
the right — cases (a), (d), and (e) — i.e., a negative phase shift, /Jo- 
Figure 5a illustrates the case of scattering far from a resonance; it is 
seen that a ^ R, /3 ^ -R/X « 1, and <r sc ^ AtvR 2 . (Figure 5 greatly 
exaggerates the magnitude of R; for slow neutrons, R, the nuclear 
radius, is ~10~ 12 cm and X is ~10 -8 cm.) This case is essentially the 
same as scattering by a rigid sphere, for which u (r < R) = 0. 

Cases (b), (c), and (d) represent various stages of scattering near and 
at resonance. It is seen that the resonance condition (case c) corre- 
sponds to a ^ ±X/4, ft, ^ ±x/2, and <r ^ 4ttX 2 . 

Case (e) is supposed to represent the class of scatterings by a finite 
potential well, rather than by a rigid sphere (e.g., neutron-proton scat- 
tering). Under such circumstances, the scattering length depends in 
detail on the depth and range of the potential. In particular, if we con- 
sider scattering by a potential of fixed depth but continuously increasing 
range, the value of a will go through "resonances" (involving both large 
magnitude and change in sign) at certain values of R. Such "nuclear 
size resonances" have recently been invoked by a number of authors 
(F38, M40, S33) to explain large variations in the potential scattering 
of adjacent nuclei. 

For neutron energies far from a resonance, potential scattering is the 
most important scattering process. In the region of a resonance, cap- 
ture scattering predominates. However, for the scattering of a neutron 
of given I, the capture and potential scattering are coherent and inter- 
fere with each other. Thus, the Breit-Wigner formula must, in the case 
of elastic scattering, be modified to 



a(n,n) = 7rX 2 /j 



i?ni 



e 2i c — 1 



2 



+ <r/ (51) 



(E - E r ) + iT/2 
where <r p ' is the incoherent part of the potential scattering, 1 

a p ' = a p - tX% e 2iit - 1 | 2 (49b) 

For most observed neutron resonances, the neutron kinetic energy is 
sufficiently low so that we have only to concern ourselves with I = 
interactions. Furthermore, except for the light nuclei (A ^ 25), the 

1 Note that the usual coefficient (21 + 1) is contained in the statistical factor /;. 



Sec. 2B] Interaction with Nuclei 273 

only reactions of significance in the region of observed resonances are 
(n,ri) and (n,y). For such resonances (with kR <<C 1), 



r = r» + r r 




<r(n,y) = 7rX 2 / - 




-L n* y 


(E- 


- E r ) 2 + T 2 /4: 


<r(n,ri) = irX 2 fo 




r„ 


(E 


- E r ) + tT/2 



(52) 



+ 2kR 



+ 4wR 2 (l - /„) 



The radiative capture resonance is of the symmetrical Breit-Wigner 
form (Eqs. 43 and 44). Because of the interference between capture 
and potential scattering, the scattering resonance is not symmetrical 
about E r . Instead, it goes through a minimum at an energy below E r 
and falls off less rapidly on the high-energy side. The position of the 
scattering minimum is given by 

r, r„x D 

*'-**- a* -WW* (53) 

and the cross section at the minimum is 

<rmin(n,n) ^ 4xE 2 ( 1 - /o + -^— j- ) (54) 

(assuming T n X/R <<C T and using Eqs. 37 and 52). Figures 6a and 6b 
show typical shapes of scattering resonances for the case of target nuclei 
of spin and spin •§• (indium). For the latter case, it must be kept in 
mind that the capture scattering involves a definite, single J value of 
the compound nucleus, while the potential scattering occurs for both 
possible J values of the system (in this case J = -§• ± ^ = 5 or 4), re- 
sulting in a finite scattering at the minimum. In addition, Fig. 6b cor- 
responds to a case where r « T y is> r„, and the presence of the radiative 
capture process also prevents the scattering minimum from going to zero. 
Another situation of interest is when r ~ T n i ^> T y , which is fre- 
quently encountered in nuclei of A < 50 for intermediate-energy neu- 
trons. In this case, application of Eq. (51) gives, for the position of the 
scattering minimum, 

r 

E r - E min = - ctn fc (53a) 

and, for the minimum cross section, 

(Tmm(n,n) = a-p (54a) 



274 The Neutron [Pt. VII 

In addition, the maximum of the resonance is shifted toward higher 
energies: 

r 

#m a x - E r = - tan fc (55) 

and the peak cross section is 

<7 (n,n) = 4:irX 2 fi + <r p ' (55a) 

For a scattering resonance involving a target nucleus of 7 = and S 
scattering only (&> « £o) the scattering cross section will be zero at 




Fig. 6a. Schematic drawing of the cross section through a slow-neutron scattering 

resonance for I = neutrons, for a target nucleus of spin 0. A spin / ^ of the 

target nucleus would change this curve by multiplying it by /o(J) and adding to it 

a constant cross section [1 — fo(J)]4:irR 2 . 

the minimum (Fig. 6a). For 7 > or I > or both, the minimum will 
not fall to zero, since the potential scattering for angular momenta not 
associated with the resonance does not interfere with the capture 
scattering. 

Returning, now, to slow-neutron reactions in heavy nuclei (Eqs. 52), 
we note that the properties of a given resonance are completely deter- 
mined in terms of four parameters, E r , Y n , T y , and / . These can, in 
turn, be ascertained from four measured quantities: (1) the resonance 
energy, E r ; (2) the resonance width, T = T n + T 7 ; and (3), (4) the 
peak capture (or total) and scattering cross sections, <To(n,y) and <r (n,n). 
(We assume that R is given by r A Va X 10~ 13 cm.) Of these, c {n,n) is 
usually most difficult to measure, since (for r 7 y> r„) it is only a small 
fraction of the total. Thus, in a study of the 0.176-ev Cd 113 (7 = ^) 
resonance, Beeman (B12) has been able to extract all the parameters, 
and to show that the level of the compound nucleus has the spin J = 1. 



Sec. 2B] 



Interaction with Nuclei 



275 



In the absence of a measurement of o- ( n , n ), it is only possible to deter- 
mine the product / r„. 

For pure scattering resonances (r « r„ 55> T y ) the statistical factor 
/ is immediately determined from the difference between the peak cross 
section and the minimum cross section, since c$ — tr m i n = 4irX 2 f. 

20 



- 


I 


- 


yur- J e = 4 


- 


J c =5 -A 


~ 


' ' ^^s. 


^= 4 -J 




Peak Values of (T sc are 
764 x 10" 24 cm 2 (,/„=• 4) and 
626xl0" 24 cra 2 (,/ c = 5). 
Resonance E r =1.44 ev. 


Jl 


J c =5 


1 1 


1 1 1 i 



.90 



2.70 



E (ev) 



Fig. 6b. The scattering cross section of indium (/ = %) for the 1.44 = ev reso- 
nance. The curves are computed from the known constants of the (capture) reso- 
nance and for the two possible values of J = 5 or 4. From Feshbach, Peaslee, and 

Weisskopf (F27). 

So far, we have discussed the case of a single resonance, neglecting the 
effects of all the other resonances. This approximation is valid in the 
vicinity of a single resonance when the level spacing D is large as com- 
pared to the level width r. If this is not the case, the levels can interfere 
with each other, provided they belong to the same class (same I, J, and 
parity). 1 Interference between levels has been discussed by a number 
of authors (B25, B24, P10, B61, S37, W29). 

Assuming, still, that there is only one resonance affecting the energy 
region in which we are interested, the cross sections far from the reso- 

1 This statement applies only to the total cross section. Interference effects can 
always occur in the angular distribution of the reaction products provided, only, that 
the levels overlap. 



276 The Neutron [Pt. VII 

nance, | E — E r | » r, are 

<r{n,a) = (56) 

(E - E r f 

a(n,n) ^ 4tR 2 (57) 

These are of especial interest in the thermal neutron energy region when 
the first resonance falls appreciably above or below thermal energies 
(I E r I ^> E fh ). In this case 

,rX 2 /r K r g 

<rth(n,a) = — — — (58) 

Recalling that X ~ E~ Yi and, from Eq. (37), that r„ ~ E*, we have, 
provided that r„ is constant over the range of thermal energies, 






(59) 



where <r* is the cross section at a specific (arbitrary) energy E* (velocity 
v*). This is the well-known l/v law for slow-neutron reaction cross sec- 
tions far from a resonance. Deviations from the l/v law will result from 
too close proximity to a resonance; Eq. (56) shows that the effect of a 
close positive energy resonance (E r > 0) is that the cross section falls 
less rapidly than l/v, while a too-close negative resonance results in a 
thermal neutron reaction cross section which falls faster than l/v. 

The thermal neutron scattering cross section will, on the other hand, 
be constant if the first resonance is sufficiently far removed. However, in 
the case of scattering, the effects of interference between the potential 
scattering and the resonance will extend to considerable distances 
(Figs. 6a and 6b), so that even a far-away resonance can have an appre- 
ciable effect on the thermal neutron scattering. 

At the other extreme, we are frequently concerned with observations 
of cross sections which represent an average over many energy levels; 
this will be the case when the energy spread of the neutrons, used in the 
measurement, is large compared to the level spacing of the compound 
nucleus at the excitation energy involved. Assuming that X, r„, r are 
essentially constant over the energy spread of the neutron source, the 
average over Eq. (44) and sum over J = I ± % yield 

2x 2 X 2 r„r„ 



a(n,a) = — (60) 

If states of I > are possible, the above average must be summed over 
all possible I values, each with its appropriate fi and appropriate neutron 
width T n i and level spacing D;. 



Sec. 2B] Interaction with Nuclei 277 

If we consider heavy nuclei and 1 = interactions only, there are two 
regions of interest: (1) r « r r » r„; here, as for thermal neutron cap- 
ture, <r(n,y) oc 1/v; (2) r ~ V n » Y y ; in this range, <r{n,y) oc 1/E. The 
total cross section, however, has a more complicated energy dependence, 
since it is now necessary to take into account the potential scattering 
as well as the capture scattering and radiative capture (F28). 

The more general approach developed by Weisskopf and co-workers 
(F27, F28, W17, B34) gives, for the cross section for the formation of 
the compound nucleus, 

<r c « x(X + R) 2 P (61) 

The factor 7r(X + R) 2 is a measure of the area of interaction of the neu- 
tron and the target nucleus; P is the probability that the neutron will 
penetrate the nuclear surface. In the resonance region, P is a sharply 
varying function of the neutron energy, being the resonance factor. In 
terms of the simple picture which we have used, the incident neutron 
wave, of wavelength X = 1/k, must join smoothly on to the neutron 
wave, A = 1/K, at the surface of the nucleus. Since (for slow and 
intermediate neutrons) K^> k, the amplitude of the neutron wave in- 
side the nucleus will, in general, be very small, except for certain ex- 
ceptional values of the neutron energy at which the amplitude of the 
incoming wave at the nuclear surface has the maximum value. These 
special cases (energies) correspond to the resonances, for which the 
joining of the wave functions was schematically represented in Fig. 5. 

We have previously (Eq. 21) given an expression for P which can be 
interpreted as the average (over many resonances) probability for the 
neutron to penetrate the target nucleus. Hence, the average reaction 
cross section is 

_ 4tt(X + R) 2 kK 

c c ^ — - V- (62) 

(k + K) 2 

For slow and intermediate neutron energies, X = 1/k » R, K = K >5> k, 

whence 

_ 4irX 2 & 4ir 500 , v 

<T C = = ~ — rr barns (62a) 

K kK E* 

(for E in electron volts). Thus, the average cross section for the forma- 
tion of the compound nucleus also follows a 1/v law in these energy 
ranges. Introducing the connection between neutron energy and neu- 
tron width from Eq. (37), 

irT n K 

k = (37a) 

2D 



278 The Neutron [Pt. VII 

the average cross section for the formation of the compound nucleus 
becomes 

- 2x 2 x 2 r n / ^ 

«*= ■ (63) 



Since the cross section a(n,a) = <r c T a /T, Eqs. (63) and (60) are seen to 
be equivalent, thereby confirming the interpretation of P (Eq. 21) as 
the average over many resonances of the penetrability. 

4. Summary ; Energy Dependence of Neutron Cross Sections. In the 

preceding discussion we have outlined the basic considerations which 
determine the interactions of neutrons with nuclei. We have observed 
that the reactions of neutrons of a given energy with a given nucleus 
depend on the level structure of the compound nucleus, at the excita- 
tion energy resulting from the neutron capture. Although we are, at 
the present state of our knowledge, incapable of predicting the details 
of the nuclear level structure, a great deal can, nevertheless, be said 
concerning the average behavior of nuclei. 

In referring to neutron energies, we shall continue to use the classifi- 
cation of Section 1B7 : 

I. Slow neutrons: E < 1000 ev 
II. Intermediate neutrons: 1 kev < E < 0.5 Mev 

III. Fast neutrons: 0.5 Mev < E < 10 Mev 

IV. Very fast neutrons: 10 Mev < E < 50 Mev 
V. Ultrafast neutrons: 50 Mev < E 

In addition to the dependence of nuclear level properties on the 
kinetic energy of the incident neutron, the nuclear level spacing is very 
different for nuclei of widely different atomic weight, A. It is therefore 
convenient to classify nuclei according to the following system : 

I. Light nuclei : A < 25 
II. Medium nuclei: 25 < A < 80 
III. Heavy nuclei: 80 < A 

These dividing lines are, of course, not to be considered sharp bound- 
aries. 

While we shall return, in a subsequent section, to the experimental 
evidence on the question of the compound nucleus level spacing as a 
function of the nuclear atomic weight, it is useful to bear in mind the 
following summary of the average level spacing of the compound nucleus : 

(1) Light Nuclei. For slow and intermediate neutrons, the levels are 
very widely spaced, being ~1-0.1 Mev apart, on the average. Thus, 



Sec. 2BJ Interaction with Nuclei 279 

very few levels are encountered, and these are relatively broad and 
easily resolved. In the fast and very fast neutron energy region, the 
levels are closer, but still separate and resolvable. 

(2) Medium Nuclei. For slow and intermediate neutrons, the levels 
are ~100-1 kev apart. Hence, few levels are encountered in the slow- 
neutron range, but many in the intermediate region. For fast neutrons 
the level spacing rapidly decreases, and the levels begin to overlap. For 
very fast neutrons the compound nucleus has a continuum of levels. 

(3) Heavy Nuclei. In the slow and intermediate neutron ranges, the 
level spacing is <~1000-5 ev. Many levels are observed for slow neu- 
trons; the levels are no longer resolved in the intermediate range; for 
fast and very fast neutrons the compound nucleus has a continuum of 
levels. 

For ultrafast neutrons and for all nuclei, the compound nucleus pic- 
ture requires considerable modification. 

(a) The Total Cross Section, c t . Most neutron cross section measure- 
ments involve the determination of the fraction of a given neutron beam 
transmitted through a known thickness of material; such measurements 
usually determine the total cross section, the sum of the cross sections 
for all processes (absorption, scattering) which remove neutrons from 
the beam, 

at = 2 <r(n,a) (64) 

a 

For a neutron flux of <j> (neutrons • cm -2 • sec -1 ) incident on a slab of 
material of density N (nuclei/cm 3 ) and thickness <(cm), the emerging, 
undeviated flux <$> is 

<j> = 4>o exp (-Nta t ) (65) 

Figure 7 shows typical slow-neutron total cross sections for the three 
categories of elements. The cross section of carbon exhibits thermal 
neutron interference effects due to the crystal structure of the graphite 
used in the measurement. Above ~1 ev, the cross section becomes and 
remains constant. The cross section of cobalt shows a scattering reso- 
nance at 120 ev. The cross section in the thermal neutron region has a 
1/v component due to radiative capture. The indium and iridium cross 
sections show many resonances and large 1/v capture cross sections for 
thermal neutrons. Owing to the close spacing of the levels, the neutron 
width is small (Eq. 37) compared to the radiation width, so that radia- 
tive capture predominates in the observed resonances. 

For intermediate and fast neutrons individual levels have been well 
resolved only in light and medium nuclei; for these relatively large level 
spacings, the neutron widths are much larger than the gamma-ray 



280 



The Neutron 



[Pt. VII 



e 3 

1 



c 
























































































<irai 


shite 
























- 








1 




























Sy 


mbol on Reference No. 
^igure in Legend 
• 1 












1 






























1 




















® 3 

1 1 l 4 l 1 






































I 4 








I A 






2 4 




2 4 7 




I 4 7 


2 4 



0.01 



E„(av) 



100 



(1) Columbia Velocity Selector (unpublished). (2) W. W. Havens, Jr., and L. J. Rainwater, Phys. Rev., 
75, 1296 (1949). (3) C. T. Hibdon and C. O. Muehlhause, Phys. Rev., 76, 100 (1949). (4) W. B. Jones, 

Jr., Phys. Rev., 74, 364 (1948). 

(a) 



100 

70 

^ 40 

en 

| 20 

13 10 

7 



























T 












Co 
























O r*J 


2,500 t 
B.OOl) fc 


arns 














Symbol o 
Figure 

• 

o 
® 


i Reference 
in Legerj 
1 
2 


No. 
d 






1 


\ 
























1 




















3 










/ 


\ 
































■I 




\ 


\ 








/ 




""^. 


->.... 


<A 


•iu^. 










y 


J 






\ 






/ 


' 














-.-.„ 




, r i 










\ 






/ 
































\ 


- — ' 


/ 













































































100 



7 2 

1000 4000 



E n (ev) 



(1) Wu, Rainwater, and Havens, Phys. Rev., 71, 174 (1947). (2) C. T. Hibdon and C. O. Muehlhause, 
Phys. Rev., 76, 100 (1949). (3) F. G. P. Seidl, Phys. Rev., 75, 1508 (1949). 

(b) 

Fig. 7. Typical total cross section vs. energy curves for slow neutrons, from 
Adair (A2). (a) Carbon, a light element. In this case, a t = <r sc - The variations, 
below ~0.04 ev, are interference effects due to the polycrystalline structure of the 
sample (see Section 5). (b) Cobalt, a medium element. Again, at ~ ct sc , except in 
the thermal neutron range. Effects of scattering resonances at 120 ev and >2000 

ev are evident. 



Sec. 2B] 



Interaction with Nuclei 



281 



20,000 
10,000 


















































































In 




































2 
1000 
























Symbol on Reference No. 
Figure in Legend 
1 

2 

3 

® 4 
o 5 












































1 








4 

<A 


















f 






















i 


, 










■1&T 


V. 


"-. 






../ 


I 




1 1 




















^100 
b" 7 

4 
2 

10 

7 

4 
2 
1 










■". 










FT' - 


■ 












































































I'l J 


£ 




































]aj 


I 


v. 


■ — i 










































u 




































































































_ 

















4 7 2 

0.1 



2 4 7 2 

1 10 

B« (ev) 



7 2 

100 



7 
1000 



(1) Borst, Ulrich, Osborne, and Hasbrouck, Phys. Rev., 70, 557 (1946). (2) Havens, Wu, Rainwater, 
and Meaker, Phys. Rev., 71, 165 (1947). (3) B. D. McDaniel, Phys. Rev., 70, 832 (1946). (4) E. Fermi 
and L. Marshall (unpublished). (5) C. T. Hibdon and C. O. Muehlhause, Phys. Rev., 76, 100 (1949). 

(0) 



4000 


Ir 














j\ 








1 1 1 1 1 1 




2 














11' 


J 'l 








Symbol on Reference No. 
Figure in Legend 




1000 
















-tr 








• l 

x 2 






7 




... 










/ 


1 












" 2 

S3 

£ loo 


— 




?** 


*-. 






l\ 


I 
































-«** 








f 






































f 


















b ' 


— 














• 




P- 






'■^- 


~~ S~ 


























. 


V 






U- 
















in 
















i 
f 























0.01 



4 7 2 

0.1 



7 2 

1 10 

.E„(ev) 



7 2 
100 



(1) Rainwater, Havens, Wu, and Dunning, Phys. Rev., 71, 65 (1947). (2) Sawyer, Wollan, Bernstein, 
and Peterson, Phys. Rev., 72, 109 (1947). (3) Powers, Goldsmith, Beyer, and Dunning, Phys. Rev., 

53, 947 (1938). 

(d) 

Fig. 7 (Continued), (c) Indium, a heavy element. The famous resonance, at 1.44 
ev, is well resolved. There are probably many unresolved resonances above 10 ev. 
(d) Iridium, a heavy element. The first two capture resonances are fairly well re- 
solved. There are probably many unresolved resonances above 20 ev. 



282 



The Neutron 



[Pt. VII 



widths, and the observed resonances are due to scattering. We post- 
pone the discussion of scattering resonances and consider, instead, the 
average behavior of the total cross section for intermediate and fast 
neutrons. The average total cross section, according to the theory of 
Feshbach and Weisskopf (Eqs. 49, 61, 62), depends on the properties of 
the nucleus involved only through the nuclear radius, R. 1 



ft! 



2.0 



1.0 



-1 1 


















" Uo = B 


















-.1 




















4 




















- y 




















_\ 




















- 


As 

V 




s 2 *" 

\ 


cje+i 


) 2 










- 




<^ 






ii; 


^ _ 








— 




















- 





















123456789 10 
x — kR 

Fig. 8. Total cross sections (averaged over resonances) as a function of x = kR, 
for Xo = KqR = 5 and 8. <ro is the total cross section for an impenetrable sphere 
of radius R. The broken curve gives the approximate behaviors for large x. From 
Feshbach and Weisskopf (F28). 

Figure 8 is a plot of the expected average total cross section vs. neutron 
energy. The cross section is given in units of wR 2 , and the energy in 
units of 

x = kR = 0.222RE' A (66) 



where R is in units of 10~ 13 cm and E in Mev. The curves are plotted 
for two values of X = K R: X = 5 (A « 40) and X = 8 .(A ~ 160). 
The curve labeled <r is the total cross section of an infinitely repulsive 

1 The parameter K «1X 10 ~ 13 cm, the wave number of a neutron in the nucleus, 
is assumed to be the same for all nuclei; this is not strictly true, but the variations 
in Ka are expected to be small. 



Sec. 2B] Interaction with Nuclei 283 

sphere of radius R. For large values of x, the curves are approximately- 
fitted by the broken curve, 

«r« « 2t(R + X) 2 (67) 

It is important to note that, while the value of the total cross section 
approaches 2icR 2 for X « R, the deviations from this asymptotic value 
are quite appreciable, even for very fast neutrons; thus, for E = 50 
Mev, X = 0.64 X 10 -13 cm, which cannot be neglected. 

In Fig. 9 the experimental values of the total cross section are com- 
pared with the theory for a number of elements: (a) iron, A « 56; 
(b) silver, A ~ 108; (c) antimony, A « 122; and (d) lead, A « 207. 
The measurements employed relatively poor resolution, and as a result 
the averaging process was, to a large extent, performed experimentally. 
The agreement with theory is seen to be excellent, except in the case of 
antimony, where the observed cross section falls considerably below the 
theoretical value at low energies. 1 

(b) Cross Section for Formation of a Compound Nucleus, a c . The cross 
section for the formation of a compound nucleus includes all reactions, 
except elastic scattering in which the quantum state of the nucleus is 
unchanged. It is usually defined as 

<?c = O t — (T e l (68) 

While this definition does not accurately take into account the effects 
of interference between capture and potential scattering, such effects 
are essentially eliminated when the cross section is averaged over many 
resonances. 

For light and medium nuclei, a c for slow and intermediate neutrons 
is mainly due to capture scattering (except for thermal and slower neu- 
trons). For heavy nuclei, in these energy regions, a c is almost entirely 
due to radiative capture. For fast and very fast neutrons, elastic scat- 
tering contributes little to the value of <r c , the other possible reactions 
— inelastic scattering, (n,p), (n,a), (n,2n), etc. — predominating. 

The energy dependence of a c (averaged over resonances) is described 
by Eqs. (61), (62), (62a) and plotted in Fig. 10. (The method of plot- 

1 This "anomalous" type of behavior has been found, by the Wisconsin group 
(M31), to occur with unanticipated frequency. Furthermore, extension of the total 
cross section measurements to 3.2 Mev has brought to light, for the heavier ele- 
ments, the frequent existence of a broad maximum in a t which appears to move to 
higher energies with increasing mass number. These phenomena are strongly sug- 
gestive of the scattering by a finite potential, and could possibly be interpreted as 
supporting an "independent particle" model for fast neutron scattering. It is, at 
present, difficult to see how these results can be reconciled with the apparent wide- 
spread applicability of statistical models. 



284 



The Neutron 



[Pt. VII 





Fe 


















10 




n 


















_ 


















OS 
N 6 


" 


k 


/i 














4 


"I 


mi 


"i 


^ 


"~-fV 


Theory 












V 


), 












?, 






Experin 


ent 














i 


i 


t 


i 


1 


1 


1 


1 


1 



0-2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.1 

x=kR 

(a) 



t3 



Ag 




















- 




-~"T 


- _ ^- 


eory 












- 


Exp 


J 

iriment 










P^ 




rV '° sr \7 


i 


1 


I 


i 


1 


1 


1 


1 


1 


i 



0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 

x=kR 

(b) 



Q5 



Sb 



~^L 



0.2 0.4 0.6 



0.8 1.0 

x=kR 



1.2 1.4 1.6 1.8 2.0 



(C) 



Fig. 9. Comparison of the theory of Feshbach and Weisskopf (F28) with measured 
total cross sections for a number of elements. The measurements are due to Bar- 
schall and co-workers (references in F28). (a) Iron, A =< 56. (b) Silver, A « 108. 
(c) Antimony, A «= 122. In the case of antimony the observed cross section is in 
better agreement with the prediction of scattering from a rigid sphere. 



Sec. 2B] 



Interaction with Nuclei 



285 



8 


Pb 




















6 






~~~- 


— ~v_ 






. Theory 






<* 


- 










Experiments^ 










2 


i 


i 


' 


| 


1 


1 


1 


1 


1 





0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 

x=kR 

(d) 



Fig. 9 (Continued), (d) Lead, A = 207. 



Sd.b 
2.4 
2.2 
2.0 
1.8 
1.6 
1.4 
1.2 


















































































I 


Vo = 


5 
















\ 








• 




-*) 2 








V 


^0=8 








^ "^ 


- :;:: ^=c: 


^^5--. 


- — 


. 
































1.0 




^cii 



















10 



Fig. 10. Cross section for the formation of a compound nucleus vs. x = kB, accord- 
ing to Feshbach and Weisskopf (F28). The solid curves are for X = KoR = 5, 8, 
and 11. The broken curve, appropriate to large x, is <r c « -^a t « ir(iJ + X) 2 . 



286 The Neutron [Pt. VII 

ting is the same as that employed in Figs. 8 and 9.) As seen from Eq. 
(62), a c -> ir(R + X) 2 for x » X ; the deviation of <r c from w(R + X) 2 
depends on the value of x/X , being «11 percent for x = X and «4 
percent for x = 2X . For large values of x, <y c ~ \<j t and, since cap- 
ture elastic scattering is negligible for large x, a c ~ a e i; i.e., for large x, 
the total cross section divides into approximately equal parts elastic 
scattering and capture. 

C. Types of Neutron Reactions 

In the previous section, we have discussed the general properties of 
neutron reactions and the formation of the compound nucleus. Although 
it has been observed that the nature of the compound nucleus de-excita- 
tion is governed by the competition between the various possible modes 
of decay, the properties of this competition have been treated in the 
most general way. This section is devoted to the anatomy of neutron 
reactions — to the details of the competition between the various possible 
neutron reactions, as they are influenced by the properties of the nuclei 
involved and by the energy of the impinging neutrons. 

The relative probability of a given neutron reaction (n,a) — neutron 
in, particle a out — is defined in terms of the cross section <r(n,a). For a 
(pure) material containing NV nuclei in a neutron flux <£, the number of 
(n,a) reactions per second is NV<f>a-(n,a) . 

1. Scattering (n,ri). Neutron scattering is one of the two reactions 
which are energetically possible for all nuclei at all neutron energies. 
(The other is radiative capture.) The observed elastic scattering is, in 
general, the result of a superposition of the potential scattering and 
the capture scattering (Eq. 51). However, this superposition is such 
that it is coherent (addition of amplitudes, interference effects) for 
the capture scattering and that part of the potential scattering which 
involves the same neutron orbital angular momentum, Zh, and the same 
total angular momentum state, /. The remainder of the potential scat- 
tering is observed as an incoherent background (addition of cross sec- 
tions) to the scattering, as described by Eq. (51). 

(a) Slow Neutrons. In the slow-neutron energy range, the neutron 
wavelength is very much larger than any nuclear radius. Hence, neu- 
trons can react (with appreciable probability) only in the I = state. 
Thus, the potential scattering cross section is 4irR 2 (Eq. 49a), and the 
scattering cross section reduces to Eq. (52). 

In the absence of nuclear resonances, the slow-neutron (potential) 
scattering cross section would be expected to increase with atomic num- 
ber as A % . Table 5 is a compilation of observed slow-neutron scatter- 
ing cross sections together with the corresponding values of 47r.fi! 2 , com- 



Sec. 2C] 



Interaction with Nuclei 



287 



puted on the assumption R = 1A7A H X 10 -13 cm. The scattering 
cross sections, in many cases, deviate considerably from 4irR 2 , being 
both larger and smaller. These deviations are too large to be ascribed to 
fluctuations in R. An additional, noteworthy fact is that the deviations 
occur, with comparable frequency, at all values of the atomic number. 

These fluctuations in <r sc can be ascribed to the effect of a nearby 
resonance, being the result of the interference between the potential and 
the resonance scattering. A value of <r sc > 4xi2 2 indicates that the 
closest resonance is at an energy lower than that at which the cross sec- 
tion was measured (frequently E r < 0), while <r sc < 4irR 2 indicates a 
resonance at a higher energy. 

The probability that a scattering measurement, made at a given 
(arbitrary) neutron energy, will fall on a resonance is inversely propor- 
tional to the level spacing. That is, the average distance of a given 
energy from a resonance is proportional to the level spacing; thus, the 
larger the level spacing, the farther, on the average, from the closest 
level. However, the region over which interference effects are impor- 
tant is also proportional to the level spacing (Eq. 53). Hence, the 
probability that a scattering measurement will give a cross section dif- 
ferent from A-wR 2 is essentially independent of the level spacing and, 
therefore, of the atomic number of the scatterer. 



TABLE 5 
Slow Neutron Scattering Cross Sections for Free Nuclei 



Element 


At 


Csc t 

(barns) 


4irfl 2 § 
(barns) 


E || (ev) 


References If 


iH 1 




20.36 







W10, M24 


iH 2 




3.3 




th 


S33 


2 He 4 




1.4 


(0.2) 


th 


H41 


3 Li 7 




~1.5 


0.99 


th 


S33 


4 Be 9 




6.1 


1.17 
(0.8) 


th 


S33 


B B 


11, 10 


3.9 


1.33 
(1.5) 


C 


W10 


6 C 12 




4.70 


1.42 
(1.8) 


th 


W10 


7 N U 




9.96 


1.58 


10-200 


W10, M25 


8 16 




3.73 


1.70 

(2.3) 


15-1000 


W10, M25 


9 F 19 




3.3 


1.93 


0.25-40 


W10, R5 



288 



The Neutron 



[Pt. VII 



TABLE 5 (Continued) 
Slow Neuteon Scattering Ceoss Sections toe Free Nuclei 



Element 


At 


<Tsc t 

(barns) 


4irj? 2 § 
(barns) 


E || (ev) 


References If 


ioNe 


20 


2.4 


2.01 


th 


H41 


u Na 23 




3.3 


2.20 


1-800 


H58 


i 2 Mg 


24, 26, 25 


3.9 


2.28 
(2.5) 


th 


S33 


«A1» 




1.35 


2.44 

(2.7) 


th 


W10 


uSi 


28 


2.2 


2.51 


1-100 


W10, R5 


i 6 P 31 




~3.3 


2.68 


1-10 


G20 


16S 


32 


~1.2 


2.74 
(2.1) 


10-400 


A2 


nCl 


35,37 


14.2 


2.93 

(2.8) 


th 


H59 


isA 


40 


0.8 


3.17 


th 


H41 


i 9 K 


39 


~2 


3.13 


th 


S33 


2oCa 


40 


3.3 


3.18 


th 


S33 


2lSc« 




12.8 


3.44 


th 


H40 


2 2 Ti 


48 


6 


3.58 


th 


S33, F27, G12 


23V 61 




5.02 


3.73 


th 


H26 


24Cr 


52 


3.7 


3.78 


th 


S33 


24Cr 63 




8.4 


3.83 


th 


H40 


26 Mn 56 




2.1 


3.93 


th 


S33 


26 Fe 54 




2.4 


3.88 


th 


S33 


2 6 Fe 56 




12.5 


3.97 


th 


S33 


26 Fe 67 




2 


4.02 


th 


S33 


2eFe 


56 


11.3 


3.97 


th 


S33 






11.1 


(3.9) 


1.44 


F27, H36 


27C0 69 




5 


4.12 


C 


F27, B57 


28 Ni 68 




27 


4.07 


th 


S33 


28Ni 6 ° 




1.0 


4.16 


th 


S33 


28 Ni 62 




9 


4.25 


th 


S33 


28 Ni 


58, 60 


16.7 


4.10 


th 


S33 


29CU 


63, 65 


7.6 


4.33 


th 


S33 






8.3 


(3.8) 


1.44 


F27, H36 


3oZn 


64, 66, 68 


4.1 


4.41 

(4.4) 


th, 1.44 


S33, F27, H36 


aoZn 67 




7 


4.48 


th 


H40 


3iGa 


69, 71 


3 


4.60 


th 


W10 


32Ge 


74, 72, 70 


8.3 


4.71 


th 


S33 


33AS 76 




~7 


4.83 


th 


S33 



Sec. 2C] 



Interaction with Nuclei 



289 



TABLE 5 (Continued) 
Slow Neutron Scattering Cross Sections for Free Nuclei 



Element 


A t 


08C X 

(barns) 


4ttK 2 § 
(barns) 


E || (ev) 


References f 










34 Se 


80, 78 


10 


5.00 
(5.0) 


C 


F27, G12 


ssBr 


79, 81 


5.9 


5.04 


th 


S33 


3eKr 


84, 86, 82, 83 


7.2 


5.20 


th 


H41 


37 Rb 


85,87 


5.4 


5.27 


th 


S33 


3sSr 


88 


9.3 


5.36 


th 


S33 


39Y 89 




3.9 


5.41 


th 


H40 


40 Zr 


90, 94, 92, 91 


~7 


5.50 


th 


S33 


4lNb 93 




6.1 


5.58 


th 


S33 


42 Mo 


98, 96, 95, 92 


7.3 


5.69 


th 


S33 


44R11 


102, 104, 101, 
99, 100 


6 


5.92 


C 


F27, G12 


4 5 Rh 103 




3.5 


5.96 


th 


H40 


46 Pd 


106, 108, 105, 


4.7 


6.11 


th 


S33 




110 


4.4 




C 


F27, G12 


4 7 Ag 107 




10 


6.12 


th 


S33 


47Ag 109 




6 


6.20 


th 


S33 


4?Ag 


107, 109 


7 


6.15 

(5.8) 


th 


S33 


4 8 Cd 


114, 112, 111, 
110, 113 


5.3 


6.32 

(6.5) 


5-100 


G20 


49ln 


115 


2.2 


6.42 


th 


W10 


soSn 


120, 118, 116 


4.8 


6.56 
(6.9) 


th 


S33 


6 iSb 


121, 123 


4.1 


6.67 

(6.7) 


th, 0.1-2 


S33, G20 


5 2 Te 


130, 128, 126 


5 


6.88 


C 


G12 


53I 127 




3.7 


6.86 


th 


S33 


5 4 Xe 


132, 129, 131, 
134 


4.3 


7.02 


th 


H41 


65 Ce 133 




~7 


7.08 


th 


S33 


56Ba 


138, 137 


8 


7.23 


C 


G12 


69 Pr 141 




7.9 


7.35 


th 


H40 


62Sm 


152, 154, 147, 
149, 148 


23 


7.68 


th 


H40 


63EU 


153, 151 


29.7 


7.73 


th 


H40 


64Gd 


158, 160, 156, 
157, 155 


26 


7.90 


th 


H40 



290 



The Neutron 



[Pt. VII 



TABLE 5 (Continued) 
Slow Neutron Scatteeing Cross Sections for Free Nuclei 



Element 


A\ 


f«c t 

(barns) 


4xfl 2 § 
(barns) 


E || (ev) 


References H 


72H1 


180, 178, 177, 
179 


25.7 


8.61 


th 


H40 


73 Ta 181 




6.9 


8.69 


th 


S33 


74W 


184, 186, 182, 
183 


5.6 


8.78 


th 


S33 


74 W 186 




23 


8.84 


th 


H40 


76 0s 


192, 190, 189, 
188 


~10 


8.98 


C 


G20, G12 


7sPt 


195, 194, 196 


11.1 


9.14 


th 


S33 


79 AU 197 




~9 


9.19 
(7.1) 


th 


S33 


soHg 


202, 200, 199, 


15 


9.30 


0.1-10 


G20 




201, 198 


21.5 


(8.7) 


th 


H60 


81T1 


205, 203 


9.6 


9.42 


0.1-1 


F27, G20 


82 Pb 


208, 206, 207 


11.5 


9.51 


th 


S33 






12.4 


(7.6) 


1-10 


F27, G20 


83 Bi 209 




9.2 


9.56 


5-10 


F27, G20 






10 


(7.8) 


th 


S33 


92U 


238 


8.2 


10.5 


th 


U2 



f When more than one isotope is involved, this column lists the most impor- 
tant isotopes, of relative abundance greater than 10 percent, in the order of 
abundance. 

t The cross sections are for free nuclei. Most observations have been made on 
bound nuclei; these have been corrected according to the relationship 



°"free 



-C-*i)' 



^bound 



§ Measured nuclear radii, taken from the collection of Blatt and Weisskopf 
(B34), are given in parentheses. Otherwise they are computed on the basis of 

R = 1.474H X 10- 13 cm 

|| The symbols have the following meanings: th, a thermal Maxwell distribu- 
tion of neutrons at a temperature of <~300°K; C, those neutrons, in a thermal 
distribution, capable of penetrating through appreciable thicknesses of cadmium. 
(These have energies above the cadmium cut-off, ~0.5 ev.) 

f This compilation leans heavily on the previous compilations of Shull and 
Wollan (S33), Way at al. (W10), and Feshbach, Peaslee, and Weisskopf (F27), 
in which references are given to the original investigations. 



Sec. 2C] 



Interaction with Nuclei 



291 



However, the curves of c sc vs. E will be very different for nuclei of 
widely different atomic number, since the number of resonance and 
their characteristics depend strongly on the level spacing. 

(1) Light Nuclei. In general, the main reaction of slow neutrons 
with light nuclei is elastic scattering. There are four exceptions — Li 6 
and B 10 , for which the main reactions are (n,a); He 3 and N 14 , for 
which there is an appreciable slow-neutron (n,p) cross section. Aside 
from these, a sc = v t (see Fig. 7a). 

(2) Medium Nuclei. With regard to the scattering of slow neutrons, 

medium nuclei are similar to light nuclei, except that resonances are 

sometimes encountered. These are primarily scattering resonances 

since the level spacing is quite large. For this case r ~ r n > r T , the 

peak cross section 

2.6/ 

<r (n,n) ^ 47rX r 2 / = X 10 6 barns (55a') 

E r 

(E r is in electron volts.) The 120-ev resonance in cobalt (Fig. 7b) is an 
example of such a scattering resonance. The peak cross section for this 
resonance should be ~13,000 barns since, for Co 59 , I = f-(/~ f). 
The smallness of the peak cross section in Fig. 7b, as well as the absence 
of the expected interference between the resonance and potential scat- 
tering, is due to the poor resolution of the measurement. Since the 

TABLE 6 
Properties of Some Neutron Scattering Resonances 













Com- 








Target 
Nucleus 


Spin 
/ 


E r (ev) 


r„ (ev) 


r„/r 


pound 
Nucleus 
Spin J 


Observed 
<r (barns) 


4xX r 2 / 
(barns) 


References 


nNa 23 


3 
2 


3,300 


340 


~0.999 


2 


550 


540 


H58, t 


16 S 32 





111,000 


18,000 


~l 


1 
2 


21.8 


23.4 


Al 


17C1 36 


2 


-75 


2.63 fl 


0.90 t 


(2) 






H59 


2 6 Mn 55 




345 


20 


0.990 


3 


4-5,000 


4,400 


S17, H40, H43 


27C0 59 


7 
2 


123 


»4 


0.94 


(4) 


12,500 


12,200 


H38, S17, H40, t 


33AS 76 


3 


46 


0.11 


0.72 






~28,000 


t 


69ft 141 


5 
1 


(-10?) 




0.93 






(~130,000) 


H40 


6 2 Sm 152 





8.2 


~0.3 


0.66 


1 
2 




320,000 


H40 


74W 186 





19.25 


0.25 


0.62 


1 
2 


~90,000 


135,000 


H39, H40, S19 


8lT1 eon) 


1 


260 


«3.2 


0.52 


(1) 




7,500 


H40 


83Bi 2M 


"Z 


770 


3.5 


~1 






~1,700 


t 



t Some of the original data in this table have been revised in the light of recent measurements by the Harwell time- 
of-flight velocity selector group (M26) and by the Argonne fast chopper group (whose members included L. M. Bol- 
linger, R. R. Palmer, and S. P. Harris). We are grateful to these groups for private communications of their results. 

t These values are at E = \ E r \ =75 ev. 



292 



The Neutron 



[Pt. VII 



recognition, by Goldhaber and Yalow (G14), that the resonance of 
manganese at ~300 ev is a scattering resonance, many such resonances 
have been observed and investigated in medium (and some heavy) 
nuclei. Table 6 summarizes the properties of a number of scattering 
resonances. 

(3) Heavy Nuclei. In heavy nuclei, with their relatively small level 
spacings, the gamma-ray width is generally larger than the neutron 
























Symbol o 
Figure 


1 1 

l Reference No. 
in Leo-end 
















1 












® 1 

oA 2 
















: 














- 




4 






















































1 












k~ 
















L 


«h < 


o 


i 




^ 






® 


/ 1 

// 


•V 




i 
















® 






V 


9 . 




l*C 


/ 




***•& 




w 


















A 










A 






A 










' 



0.2 0.4 0.G 0.8 1.0 1.2 1.4 1.6 1.8 

£„(Mev) 

(1) Fields, Russell, Sachs, and Wattenberg, Phys. Rev., 71, 508 (1947). (2) Adair, Barschall, Bockel- 

man, and Sala, Phys. Rev., 75, 1124 (1949). (3) Bockelman, Miller, Adair, and Barschall, Phys. Rev., 

84, 69 (1951). (4) Freier, Fulk, Lampi, and Williams, Phys. Rev., 78, 508 (1950). 

Fig. 11. Total (scattering) cross section of O 16 for intermediate and fast neutrons, 
showing well-resolved P-wave resonances at 0.440, 1, and 1.3 Mev; from Adair 
(A2). Measurements from ~1.8 to 4 Mev have exhibited additional resonances 
whose angular momenta and parities have been deduced by Baldinger, Huber, and 
Proctor (B5) by means of accurate measurements of the angular distributions of the 
elastically scattered neutrons. 

width, r ~ T 7 > r„. Accordingly, the peak scattering cross section is 



small compared to the peak radiative (and total) cross section, 



<ro(n,n) = ~ <r (n,y) 



(47a) 



The resonance scattering of cadmium (B12) is an example of this case. 
Owing to fluctuations in the nuclear level spacing, and to the differ- 
ences in neutron binding energies, levels are sometimes encountered in 
heavy nuclei for which T n > F y . The broad resonance of W 186 at 19 ev 
has been found to be such a scattering resonance (H39). 1 

1 The existence of broad, scattering resonances for slow neutrons on heavy nuclei 
might be additional evidence for the necessity, under certain circumstances, of re- 
garding neutron reactions from the independent particle point of view. 



Sec. 2C] 



Interaction with Nuclei 



293 




(su-req) % D 



u 








o 


<y 






&H 


d 


E- 




X 


E-i 



CD 

d 




'55 


(H 




T3 




fe 




d 


d 




03 










a> 






OJ 


•d 






o 




d 




d 




+-> 




oi 
(=1 


o 


d 




d 


o 




tn 

CD 




05 




ffl 


ri 




^ 


t-. 






T3 


to 
=3 




> 




.id 




O 








Eh 


d 


F-H 
























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294 



The Neutron 



[Pt. VII 




(siueq) ?x> 



Sec. 2C] 



Interaction with Nuclei 



295 



(b) Intermediate Neutrons. The intermediate neutron energy region 
is, with respect to the interaction of neutrons with nuclei, a transition 
region. At the low-energy end, the neutron wavelength is still large 



V 

i-1 




















' 






















V 


















\ 


» v 


VVi 


\\ 


A 












1 




l\ 


y v 


\-> s 


f\A' y )n 


■\h[ 


Ws 


o°A 




o 

o 


A 










A 





0.4 0.6 

£7„(Mev) 



1.0 



Fig. 13b. Scattering cross section for intermediate-energy neutrons of V 51 ; data 
from Blair and Wallace, Phys. Rev. 79, 28 (1950). 



S 3 



s 




n 


n 










Symbol o 
Figure 


n Reference N 
in Legend 
1 








•1 


1 


If 










J 


L 


• 




3 


/ 

/ 
/ 


i 




'1 




\ 


\i 


'■Ahh 


{ 






Ute 






i 


/ 


v\ 




« 








\i 










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A 




I 








A 







0.7 0.E 



1.2 1.4 

E„(Mev) 



1.6 



1.8 



(1) Peterson, Barschall, and Bockelman, Phys. Rev., 79, 593 (1950). (2) Freier, Fulk, Lampi, and 
Williams, Phys. Rev., 78, 508 (1950). (3) Fields, Russell, Sachs, and Wattenberg, Phys. Rev., 71, 

508 (1947). 

Fig. 13c. Scattering cross section of S 32 in the fast-neutron region; from Adair (A2). 

compared to nuclear radii ; at the upper end, the neutron wavelength is 
comparable with (and, for heavy nuclei, smaller than) nuclear dimen- 
sions. In this region, the character of the neutron scattering process 
slowly changes. 

(1) Light Nuclei. Figure 11 shows the scattering cross section of 
oxygen (7 = 0) in the intermediate (and lower end of the fast) energy 
range. The cross section is constant up to the resonance at 440 kev. 



296 The Neutron [Pt. VII 

This resonance must be ascribed to capture of I = 1 neutrons (P reso- 
nance), since its peak cross section (14 barns) is greater than 47rX r 2 = 6 
barns, and since it shows none of the interference properties associated 
with <S resonances. The value of the peak cross section is consistent 
with a value J = -J (/i = 2) for the compound nucleus; i.e., a a (n,n) 
= 87rX r 2 + <r p ' = 12 + 4 = 16 barns. The resolution of the measure- 
ment is excellent (~10 kev), but its finite value is sufficient to account 
for the difference of ~2 barns between the observed and expected peak 
cross sections. 

(2) Medium Nuclei. In Fig. 12 is plotted the total (scattering) cross 
section for sulfur {A = 32). This curve is especially interesting, since 
it illustrates many of the features of scattering resonances which we 
have previously discussed. The broken curve is a plot of 4xX 2 . The 
three *S-wave resonances, at 111 kev, 375 kev, and 700 kev, have peak 
cross sections close to 4ttX 2 , indicating the excellent resolution of the 
measurements. They show very well the interference between the po- 
tential and resonance scattering. The other six resonances appear to be 
due to P(l — 1) or D(l = 2) neutrons, and are incompletely resolved. 
Of these, the 585-kev resonance can be ascribed to a / = f- state of the 
compound nucleus, S 33 (P13). 

(3) Heavy Nuclei. For heavy nuclei in the intermediate-energy re- 
gion, the resonances are usually much more closely spaced than the 
resolution of the best available measuring devices. At the low-energy 
end the resonances are still primarily due to radiative capture (T y > T n ), 
while at the upper end the neutron width has caught up with the gamma- 
ray width, and scattering predominates. Toward the high-energy end 
of the intermediate region inelastic scattering starts to compete as a 
possible reaction, but we postpone the discussion of this reaction. 

For certain of the heavy nuclei, the level spacing of the compound 
nucleus, at the excitation energy corresponding to neutron capture, is 
anomalously large. Such appears to be the case for the magic number 
or near magic elements lead and bismuth, owing either to an anomalously 
small neutron binding energy, or to an intrinsically large level spacing, or 
both. The cross section of Pb 206 is shown in Fig. 13a. As an interesting 
comparison, we also plot the cross section of V 51 in the intermediate 
region (Fig. 13b), and that of S 32 in the fast-neutron region (Fig. 13c). 
All of these show approximately the same level spacings. 

The curves of Fig. 13 can be used to illustrate some other interesting 
possibilities in scattering resonances. If the resonances are still resolva- 
ble at relatively large values of x = kR >" 1 (due either, as in S 32 , to 
large spacing for fast neutrons or, as in Pb 206 , to a large nuclear radius), 
I = 1 potential scattering becomes appreciable, and P-wave resonances 



Sec. 2C] Interaction with Nuclei 297 

are no longer expected to be symmetrical. Thus, the 1055-kev sulfur 
resonance must, because of the large peak cross section, be ascribed to 
neutrons of I > 0; however, it also shows a definite interference mini- 
mum, due to the P-wave potential scattering. It is therefore a Py 2 reso- 
nance. 

Another interesting anomaly occurs in the region £ = % ~ v/2. 
(&>o ~ f/2, for which the same considerations apply, requires neutrons 
of much greater energy.) For an <S-wave resonance in this energy region, 
application of Eqs. (53a) and (55) indicates that the minimum cross 
section, o- min = a p ' , occurs at E min = E r , while the maximum disappears 
(i.e., -E/max — »• °°). This leads to cross-section curves which have the 
appearance of inverted resonances, as observed in the Pb 206 cross sec- 
tion of Fig. 13a. 

(c) Fast Neutrons. Of the total cross section for fast neutrons, ap- 
proximately half involves the formation of a compound nucleus; the 
rest is due to elastic (potential) scattering, in which the neutron passes 
close to, but not into, the target nucleus. 

I. Capture Scattering and Inelastic Scattering 

With the exception of a few light nuclei (and the heaviest nuclei) the 
most probable result of fast-neutron capture is neutron re-emission. As 
long as the incident neutron energy is less than the energy of the first 
excited state of the target nucleus, the neutrons will be re-emitted with 
their incident energy (minus, of course, the energy lost to the recoil 
nucleus to conserve momentum). As soon as the incident neutron 
energy exceeds the energy required for the excitation of the lowest level 
of the target nucleus, it becomes possible for the product nucleus to be 
left in the excited state; correspondingly, the emitted neutron will have 
a smaller kinetic energy than the incident neutron. This process is 
known as inelastic scattering. 

From the point of view of the excited compound nucleus, decay to the 
ground state (capture elastic scattering) and to the various energetically 
possible excited states (inelastic scattering) of the product nucleus are 
competing processes. Each mode of decay is characterized by a partial 
width, r,-, which is a measure of its relative probability. However, from 
the experimental viewpoint, there are, in addition to the difference in 
energy of the elastically and inelastically scattered neutrons, two dis- 
tinguishing features between the two processes: (1) Capture elastic 
scattering and potential scattering are coherent processes, and the elas- 
tic scattering cross section can therefore exhibit interference effects of 
the type previously discussed. (2) Inelastic scattering leaves the prod- 



298 The Neutron [Pt. VII 

uct nucleus in an excited state, from which it decays by the emission of 
one or more gamma-rays. Indeed, it was through the observation of 
this gamma-radiation that the inelastic scattering process was dis- 
covered (Lll, K12). There have been a number of subsequent investi- 
gations of inelastic scattering by the observation of the resulting gamma- 
radiation (A29, S12, B13, G27); however, all but the most recent of 
these have been too crude to permit detailed interpretation. 

The most useful information concerning inelastic scattering has been 
derived from experiments involving the observation of the energy dis- 
tribution of inelastically scattered neutrons. The interpretation of such 
experiments has been discussed by Feld (F9). 

In the ideal inelastic scattering measurement, a monoenergetic neutron 
beam would be scattered from a thin target of the element under in- 
vestigation; the scattered neutrons would have a line spectrum, each line 
corresponding to the excitation of a given level of the target nucleus. 
The energies of the inelastically scattered neutron groups would give 
the positions of these energy levels, and the relative strengths of the 
groups would be a measure of the relative values of the partial widths 
for the corresponding compound nucleus decay. In addition, the angu- 
lar distribution of a given group of inelastically scattered neutrons (with 
respect to the direction of the incident neutrons) would yield valuable 
information concerning the angular momentum and parity properties 
of the levels involved. Finally, each group of inelastically scattered 
neutrons is accompanied (in coincidence) by one or more gamma-rays; 
the correlation between the direction of emission of the neutrons and 
gamma-rays would yield further information concerning the angular 
momentum and parity properties of the levels. 

An experiment of the type outlined above is, in practice, exceedingly 
difficult, since it requires, in addition to a monoenergetic neutron source, 
a fast-neutron detector of high resolution. Most attempts to measure 
the energy distribution of inelastically scattered neutrons have em- 
ployed "threshold" detectors, i.e., a detector sensitive only to neutrons 
of energy above a fixed value, E t . Since, in most cases, the sensitivity 
function of the detector is not well known, such experiments are usually 
difficult to interpret unambiguously. 

Many inelastic scattering measurements have employed heteroener- 
getic neutron sources, such as the Ra-a-Be or Ra-a-B source, and thresh- 
old detectors. Owing to the extreme difficulty in interpreting such ex- 
periments (S65), they will be omitted from this discussion. 

(1) Light Nuclei. The low-lying levels of light nuclei are very widely 
spaced (~0.5-5 Mev). Hence, fast neutrons can excite few levels. In 
many cases, for neutrons of a few Mev, only one level of the target 



Sec. 2C] Interaction with Nuclei 299 

nucleus will be available for excitation. The inelastic scattering cross 
section will depend in detail on the properties of this level, according to 
the following general considerations: Let <r c be the cross section for the 
formation of a compound nucleus. 1 The cross section for the excitation 
of the level (through inelastic scattering) is 

<r z = — (42a) 

Ti, the partial width for inelastic neutron emission, is given by Eqs. 
(37) to (40) : 

T x « (21 + ^CDiE^vi^R) (38a) 

in which the symbols have their usual meanings; E\ and ki are the 
energy and wave number of the (inelastically) scattered neutron : 

E^E - D x (33a) 

(E is the incident neutron energy; D x is the excitation energy of the 
target nucleus level involved.) The total width 

r = r + t 1 + r a + r 6 + • • • (28a) 

includes the width for elastic neutron re-emission, r , and the widths 
for decay by emission of any other particles, r , Tb, etc. 

In the expression for a\ (Eq. 42a) all the factors are, to a greater or 
lesser extent, energy dependent. However, in the immediate neighbor- 
hood of the threshold (E « Dx), the main energy dependence arises 
through the factor Ei A vi{kiR) in IV (It is clear that, for energetic 
reasons, Ti = for E < D x .) If we consider a region close to the 
threshold, such that E x <<C E, the factors a c and r are essentially con- 
stant. For inelastic scattering in which S-wave (I = 0) neutrons are 
emitted, the cross section will have an E^ A energy dependence near the 
threshold; for emission of I > neutrons, the cross section near thresh- 
old will increase as E^ l+y2) . 

The angular momentum of the emitted neutrons is determined by the 
values of the spins and parities of the initial (compound nucleus) and 
final (product nucleus) states. In many cases these values are such that 
neutrons of only a single, definite angular momentum can be emitted. 
In some cases, however, the spin values are such that angular momen- 
tum conservation could be satisfied in a number of alternative modes of 
decay, i.e., more than one I value is possible for the emitted neutron. 

1 The energy dependence of a c has been discussed on page 283. It should, however, 
be borne in mind that the general considerations, used to derive Fig. 10, are of ques- 
tionable validity for light nuclei in the fast-neutron range. 



300 The Neutron [Pt. VII 

In such cases the parity selection rule further limits the number of pos- 
sible I values; e.g., if the parity of the initial state is the same as that of 
the final state, neutrons can be emitted only with even I values; for 
states of opposite parity, only odd I values are permitted. Consequently, 
and also because of the strong energy dependence of Ti, one — the lowest 
— of the possible I values of emitted neutrons will be predominant in 
most cases. 

When the angular momentum of the emitted neutrons has a definite 
value and if, furthermore, the absorption of the incident neutrons to 
form the compound nucleus state involves a single I value, the inelasti- 
cally scattered neutrons will have a definite and predictable angular 
distribution with respect to the direction of the incident neutrons. The 
angular distributions, in a number of possible cases, have been given by 
Hauser and Feshbach (H44) . 

As an example of the type of angular distribution that could be en- 
countered, we give the result for a specific (hypothetical) case: 

Target nucleus: I = \ 

Compound nucleus : J = 3 
Product nucleus: /'=■§• 
Incident neutron: I = 2 
Scattered neutron: V = 1 

The resulting angular distribution of (inelastically) scattered neutrons is 

W(e) = i + |f cos 2 e 

(0 is the angle between the incident and scattered neutrons.) 

Similar information concerning the three states involved could be 
obtained by observation of the angular correlation between the inelasti- 
cally scattered neutrons and the gamma-radiation which follows the 
inelastic scattering. In the example given above, the transition AI = 1 
with parity change would be achieved through the emission of an electric 
dipole (or magnetic quadrupole) gamma-ray. Angular correlations in 
successive particle gamma-ray emission are discussed in Part IX of 
Volume III. 

As previously indicated, many cases of inelastic scattering will be 
considerably more complicated than the example discussed above. For 
instance, it may be possible for the incident neutrons to excite more 
than one compound nucleus state, requiring a number of incident 
neutron I values. Correspondingly, the decay of the compound nucleus 
to the excited product nucleus can also involve a number of different 



Sec. 2C] Interaction with Nuclei 301 

angular momenta, and the angular distributions for the different I 
values can interfere. If this situation prevails, the calculation of the 
angular distribution of the inelastically scattered neutrons requires 
further knowledge of the details of the competition. Some of these 
possibilities have been considered by Hauser and Feshbach (H44) and 
by Wolfenstein (W38) : 

There has been very little experimental investigation of the inelastic 
scattering of fast neutrons by light nuclei. The work of Beghian, Grace, 
Preston, and Halban (B13, G27), on the inelastic scattering of 2.5-Mev 
neutrons by beryllium, carbon, fluorine, magnesium, sulfur, chromium, 
iron, and copper is of interest in connection with the above considera- 
tions. They detected the inelastic scattering by observing the resulting 
gamma-radiation and used the intensity of the gamma-rays as a meas- 
ure of the cross section. In the cases of carbon and beryllium they 
were unable to detect any gamma-radiation. For the other elements 
they observed a single (monoenergetic) gamma-ray in the first four 
cases, indicating that only a single level of the target nucleus was in- 
volved; the last two yielded complex gamma-ray spectra. These results 
are summarized in Table 7a. 

The observed cross sections, when combined with a knowledge of <r c , 
yield values of the relative probability for inelastic scattering Ti/T 
(Eq. 42a). These are shown for fluorine, magnesium, and sulfur in 
Table 7b, together with the data from which they were derived. The 
values of cr c have been estimated from the available data (A2, W32). 
In the last column of the table, we also give theoretical values of Ti/T, 
based on the assumption that only elastic and inelastic scattering are 
of importance (r = r + 1^), and that both types of scattering in- 
volve only I = neutrons. 

In the case of magnesium the inelastic scattering has been observed 
directly by Little, Long, and Mandeville (L20). They scattered 2.5-Mev 
neutrons (from a D-D source) in a block of magnesium and measured 
the energy distribution of the scattered neutrons by observing proton 
recoils in a cloud chamber. In addition to the elastically scattered 
group, they observed a single group of inelastically scattered neutrons 
whose energy corresponds to an excited state of magnesium at 1.30 Mev 
(compare Table 7a). From the strengths of the two groups, they com- 
puted values of ~1.6 barns and ~0.6 barn for the elastic and inelastic 
scattering cross sections, respectively, in excellent agreement with the 
result of Grace et at. and the measured value of c t ~ 2.2 barns. The 
value of Ti/T, shown in parentheses in Table 7b, is computed from their 
value of <ti « 0.6 barn. 



302 



The Neutron 



[Pt. VII 



TABLE 7 

(a) Results of Grace, Beghian, Preston, and Halban (B13, G27) on Inelastic 
Scattering of 2.5-Mev Neutrons, from Observation of the Resulting Gamma-Radiation 



Element 


Atomic Weight 


Observed 
E y (Mev) 


Observed 
a\ (barns) 


Be 


9 




<0.014 


C 


12 




<0.006 


F 


19 


1.3 ±0.1 


0.52 ±0.18 


Mg 


24(77%), 25(12%), 
26(11%) 


1.4 ±0.1 


0.75 ±0.23 


S 


32 


2.35 ±0.15 


0.38±0.1 


Cr 


52(84%), 53(10%), 
50(4%), 54(2%) 


1.4 ±0.1 


1.2 ±0.4 


Fe 


56(92%), 54 (6%), 


0.8 ±0.1 


1.8 ± 1.3 




57(2%), 58(0.3%) 


2.2 ±0.2 


0.14 ±0.05 


Cu 


63(69%), 65(31%) 


1.1 ±0.1 


1.2 ±0.6 






2.2 ±0.1 


0.34 ±0.12 


(b) Interpretation 














Tx/r 


Ele- 


(barns) 


Refer- 
ence 


Estir 
<r c (b 


nated 
arns) 






ment 


















Experii 


nent 


Theory 


F 


2.7 


W32 


1 


.2 


0.4 




0.41 


Mg 


2.2 


G20, A2 


1 


.5 


0.5 (0 


.4) 


0.40 


S 


2.8 


G20, A2 


2 




0.2 




0.19 



(2) Medium Nuclei. In the scattering of fast neutrons by medium 
nuclei, the product nucleus can be left in any one of many (energetically) 
available excited states; correspondingly, many groups of inelastically 
scattered neutrons will be emitted by the compound nucleus. The con- 
siderations applied in the previous section for a single level can easily be 
generalized to the case of many levels. Let c; be the cross section for 
the excitation, through inelastic scattering, of the z'th level of the target 
nucleus (excitation energy Z>,) : 

<n = (42b) 



Sec. 2C] Interaction with Nuclei 303 

The partial width I\- is obtained by the substitution of the subscript i 
for 1 in Eqs. (38a) and (33a). 

Since, for intermediate nuclei and fast neutrons, neutron re-emission 
is usually predominant over all other processes, the total width is 

n 

r = r + r : +• • •+ i\ + • • •+ r n = £ r,- (28b) 

in which the first term is the width for elastic re-emission and the suc- 
ceeding terms correspond to inelastic scattering in which all possible 
levels (up to the highest for which D n < E) of the target nucleus are 
excited. 

As in the case of one-level excitation, the value and energy depend- 
ence of a specific I\- are determined by the angular momentum proper- 
ties of the emitted neutrons which, in turn, derive from the spins and 
parities of the levels involved. Since, for medium compound nuclei 
formed by the capture of fast neutrons, many levels involving many 
angular momenta are likely to be excited, decay to a given product 
nucleus state will be possible through the emission of neutrons of a 
number of different angular momenta. In the ensuing competition, 
1 = emission when possible, will usually predominate. However, for 
some levels, I = emission will be impossible, and these will usually be 
less strongly excited. For a complete description it is, of course, neces- 
sary to take into account all the possible I values of the emitted neutrons. 

Nevertheless, for purposes of illustration it is of interest to consider 
the energy dependence of the inelastic scattering cross sections under 
the assumption of <S-wave scattering only. The values of the first few 
<Ti vs. E are shown in Fig. 14; in addition to the assumption of $-wave 
scattering, we have also assumed uniform level spacing for the product 
nucleus, i.e., Z>i = D, D t = iD. The uppermost curve, labeled a oc , is 
the relative cross section for capture elastic scattering. At a given 
value of the incident neutron energy E, the total cross section for in- 
elastic scattering is 

n 

"in = z2 c i ~ °c ~ "oc (68') 

i=l 

The data on the inelastic scattering of fast neutrons by intermediate 
nuclei are meagre. Barschall, Battat, Bright, Graves, Jorgensen and 
Manley (B8) have measured the energy distribution of 3.0 and 1.5 Mev 
neutrons scattered by iron, using, as an energy-sensitive detector, pro- 
ton recoil proportional counters with different "bias" values. Although 
the resolution of these measurements was rather crude, the results can 
be satisfactorily understood in terms of the theory outlined above and 



304 



The Neutron 



[Pt. VII 



the (three) known levels of Fe 56 of excitation energy less than 3 Mev 
(F9). The results of Grace et al. (G27), shown in Table 7a, are in fair 
agreement with those of Barschall et al. for iron. 

(3) Heavy Nuclei. In the scattering of fast neutrons by heavy nuclei 
a large number of energy levels of the target nucleus are available for 
excitation. Separate groups of inelastically scattered neutrons will, in 
general, not be observed, both because of the limited resolving power of 




Fig. 14. Cross sections, <r,-, for the excitation of the low-lying levels of the target 

nucleus as a function of the incident neutron energy, E. The curves are based on 

the assumptions: (1) constant level spacing; (2) S scattering only; (3) I\ = 

C{E — Ei) Vi , with C constant over all levels. 

all fast-neutron detectors and because the separation of levels in the 
product nucleus is usually smaller than the spread in energy of avail- 
able fast-neutron sources. The inelastically scattered neutrons will 
appear to have a continuous spectrum (except, possibly, for the elasti- 
cally scattered and a few adjacent high-energy groups) ranging from 
zero to the incident energy E. 

Because of the large number of levels involved, it is no longer fruitful 
to attempt to interpret such experiments in terms of a theory involving 
the properties of individual levels. Instead, it is possible to apply a 
statistical theory, as developed by Weisskopf (W16, B34). The statisti- 
cal theory predicts an energy distribution of the inelastically scattered 
neutrons which is of the Maxwellian form 



da(e,E) ^<To— 2 e- E/r dz 



s 

Ji2 



(69) 



Sec. 2C] Interaction with Nuclei 305 

In the above, da(e,E) is the cross section for the scattering of a neutron 
of initial energy E into the energy between e and s + de. The constant 
T, referred to as the nuclear temperature, is a measure of the excitation 
of the -product nucleus after the emission of the inelastically scattered 
neutron. Strictly speaking, it is not energy independent but rather a 
function of the excitation energy of the residual nucleus, E — e. How- 
ever, for most cases of interest, T « E, and the major part of the spec- 
trum of inelastically scattered neutrons is in the energy region s <JC E; 
for this part of the spectrum, T may be regarded as essentially con- 
stant, and roughly corresponding to the full possible excitation energy 
of the product nucleus. However, for the high-energy portion of the 
spectrum, for which e ~ E, Eq. (69) is not a good approximation to 
the inelastically scattered neutron energy distribution. 

The energy dependence of Eq. (69) can be understood in terms of two 
opposing factors in the competition between the various possible modes 
of de-excitation of the compound nucleus through neutron re-emission: 
(1) The energy dependence of the neutron scattering width favors the 
emission of high-energy neutrons. This effect is responsible for the first 
factor e in Eq. (69) when proper account is taken of the emission of neu- 
trons in all possible angular momentum states. (If neutrons were emit- 
ted only in the I = state, the factor would be s H .) (2) The number of 
available levels of the product nucleus increases rapidly with the exci- 
tation energy, thus favoring the emission of low-energy neutrons. This 
effect leads to the exponential factor in Eq. (69). 

The competition between the two factors results in a maximum, in 
the scattered neutron energy distribution, at an energy intermediate 
between and E. T, the temperature of the product nucleus, is a 
measure of its level density at the excitation energy remaining after 
the emission of the inelastically scattered neutron. From statistical- 
mechanical considerations, it can be shown that (B34) 

1- d 

= [In u(E')] (70) 

T(E') dE' K J 

where w(E') is the nuclear level density at the excitation energy E' 
= E — s. For an exponential energy dependence of the nuclear level 
density, as given by Eq. (41), 

T(E') ~ (-J (70a) 

In heavy nuclei, the very rapid increase of nuclear level density with 
increasing excitation energy has the effect that the maximum of the 



306 



The Neutron 



[Pt. VII 



scattered neutron energy distribution is at relatively low energies; cor- 
respondingly, the excitation energy of the product nucleus is, for the 
major fraction of the inelastic scattering, at an energy close to the maxi- 
mum possible excitation energy, E' <~ E. [This rapid increase in level 
density corresponds to large values of the constant a (Table 4) and 



2.4 
2.0 
1.6 
1.2 
0.8 
i 0.4 



w 


(a) E 


= 3 Mev 


^- ■ 


1 


I 




, 



1.0 



1.6 
E(Mev) 



- / 


^v- (b).E=1.5 Mev 




I 




- 


^ ~ 1 



0.5 



E(Mev) 



1.0 



Fig. 15. Energy distribution of inelastically scattered neutrons from wolfram. 

The experimental results of Barschall et al. (B8) are plotted as histograms. The 

smooth curves are derived from the statistical theory and Eq. (69) (F9). 

hence to small values of the nuclear temperature, T <K E.] Thus, over 
the largest part of the spectrum, the nuclear temperature can be closely 
approximated by 



T(E) 



0" 



(70b) 



which is independent of the energy of the scattered neutrons. It is in 
this approximation that Eq. (69) is valid. 

The observations of Barschall et al. (B8), on the inelastic scattering 
of 1.5 and 3.0 Mev neutrons by wolfram (tungsten), can be interpreted 
in terms of the statistical theory and Eq. (69) (F9). The results of their 
measurements on the energy distribution of the inelastically scattered 
neutrons are plotted as histograms in Fig. 15, together with the predic- 
tions of the theory (smooth curves). The theoretical curves have been 



% (E,E t ) = C 

Jo 



Sec. 2C] Interaction with Nuclei 307 

computed on the assumption of T = 0.5 and 0.35 Mev for E = 3.0 and 
1.5 Mev, respectively, corresponding to a = 12 Mev -1 , in good agree- 
ment with Table 4. 

Table 8 is a collection of integral inelastic scattering cross sections 
and temperatures for medium and heavy nuclei. The measurements 
are represented by the cross section for the scattering of a neutron of 
incident energy E to an energy e < E t , 

dc(E,z) (69a) 

'o 

and by the temperature corresponding to the emitted neutron distribu- 
tion. Only those measurements have been included for which the inci- 
dent neutron energy and the detector threshold are relatively well 
defined. Some of these are amenable to interpretation along the lines 
outlined above in the discussion of the results for wolfram. For most, 
the energy sensitivity of the threshold detector used is not sufficiently 
well known to permit more quantitative conclusions. 

However, these measurements do give an indication of the energy 
distribution of the inelastically scattered neutrons. In particular, they 
show that, also with regard to inelastic scattering, the magic number 
nuclei lead and bismuth behave in an anomalous fashion; their scatter- 
ing is much more similar to that of medium nuclei than to that of heavy 
nuclei. The importance of this observation is that it relates to the rela- 
tively low-lying level spacings of the target nuclei, which appear, from 
these observations, to be anomalously large for lead and bismuth. The 
previously mentioned evidence on the small capture cross sections and 
absence of slow-neutron resonances for these nuclei reflect a wide level 
spacing of the compound nucleus. 

In addition to the direct observation of inelastically scattered neutrons 
or the resulting gamma-radiation, there is another method of detecting 
inelastic scattering, applicable only to certain special nuclei. These are 
nuclei which have a metastable (long half-life) level at an energy less 
than that of the incident neutrons; when excited, the metastable state 
can be detected by the resultant radioactivity. The metastable state 
of In 115 (energy 340 kev, half-life 4.5 hr) was first observed by Gold- 
haber, Hill, and Szilard (G13) as resulting from the inelastic scattering 
of fast neutrons. 

A metastable state can be induced either by direct excitation (the 
product nucleus is left in the metastable state), or indirectly, through 
the excitation, by inelastic scattering, of a higher state of the product 
nucleus, which subsequently decays (by gamma-ray emission) to the 
metastable state. In the first case, Eqs. (42a) and (38a) determine the 



308 



The Neutron 



[Pt. VII 



TABLE 8 
(a) Measured Values of Integral Inelastic Scattering Cross Sections, <n n {E,E t ) 



Ele- 
ment 


E (Mev) 


E t (Mev) 


Detec- 
tor t 


<Ti n (E,E t ) X 

(barns) 


Refer- 
ence 


<j c (barns) 
(Fig. 10) 


Be 


2.5 


2.5 


y 


<0.014 


G27 






14 


~3 


Al 


0.16 ±0.07 


P15 








~11 


Cu 


0.82 ±0.03 


P15 


0.79 


B 


14 


~3 


Al 


0.24 ±0.04 


P15 








~11 


Cu 


0.69 ±0.10 


P15 


0.64 


C 


2.5 


2.5 


7 


<0.006 


B13 






14 


~11 


Cu 


0.85 ±0.02 


P15 


0.74 


F 


2.5 


2.5 


7 


0.52 ±0.18 


G27 


1.4 


Mg 


2.5 


2.5 


7 


0.75 ±0.23 


G27 


1.10 






~2.5 


cc 


~0.6 


L20 




Al 


14 


~3 


Al 


0.62 ±0.07 


P15 








~11 


Cu 


1.06 ±0.05 


P15 


0.93 


S 


2.5 


2.5 


7 


0.38±0.1 


G27 


0.90 


Cr 


2.5 


2.5 


7 


1.2 ±0.4 


G27 


1.47 


Fe 


1.5 


~0.5 


pc 





B8 








~0.9 


pc 


0.6 


B8 


1.58 




2.5 


2.5 


7 


1.9 ± 1.3 


G27 


1.52 




3.0 


~0.75 


pc 


0.3 


B8 








~1.50 


pc 


0.7 


B8 








~2.25 


pc 


1.1 


B8 


1.50 




14 


~2 


p 


0.78 ±0.03 


P15 








~3 


Al 


1.21 ±0.03 


P15 








~11 


Cu 


1.45 ±0.02 


P15 


1.34 


Co 


1.5 


~0.5 


pc 


(0) 


B8 








~0.9 


pc 


(0.2) 


B8 








~1.3 


pc 


(0.8) 


B8 


1.62 


Ni 


1.5 


~0.5 


pc 


(0) 


B8 








~0.9 


pc 


(0.1) 


B8 








~1.3 


pc 


(0.6) 


B8 


1.62 



Sec. 2C] 



Interaction with Nuclei 



309 



TABLE 8 (Continued) 
(a) Measured Values of Integral Inelastic Scattering Cross Sections, <Ti n (E,E t ) 



Ele- 
ment 


E (Mev) 


E, (Mev) 


Detec- 
tor f 


a in (E,E t ) t 
(barns) 


Refer- 
ence 


(T c (barns) 
(Fig. 10) 


Cu 


1.5 


~0.5 


pc 


(0.3) 


B8 








~0.9 


pc 


(0.6) 


B8 








~1.3 


pc 


(0.9) 


B8 


1.65 




2.5 


2.5 


y 


1.5 ±0.7 


G27 


1.62 




3.0 


~0.75 


pc 


(0.6) 


B8 








~1.50 


pc 


(1.3) 


B8 








~2.25 


pc 


(1.5) 


B8 


1.61 


Cd 


14 


~2 


p 


1.14 ±0.04 


P15 








~3 


Al 


1.66 ±0.07 


P15 








~11 


Cu 


1.89 ±0.06 


P15 


1.85 


Ta 


1.5 


~0.5 


pc 


(1.4) 


B8 








~0.9 


pc 


(2.0) 


B8 








~1.3 


pc 


(2.7) 


B8 


2.57 


W 


1.5 


~0.5 


pc 


0.9 


B8 








~0.9 


pc 


2.1 


B8 


2.57 




3.0 


~0.75 


pc 


1.4 


B8 








~1.50 


pc 


2.4 


B8 








~2.25 


pc 


2.8 


B8 


2.63 


Au 


3.0 


~0.75 


pc 


(2.1) 


B8 








~1.50 


pc 


(2.8) 


B8 








~2.25 


pc 


(3.0) 


B8 


2.69 




14 


~2 


p 


1.47±0.10 


P15 








~3 


Al 


2.06 ±0.09 


P15 








~11 


Cu 


2.51 ±0.04 


P15 


2.69 


Pb 


1.5 


~0.5 


pc 





B8 








~0.9 


pc 


0.4 


B8 


2.81 




2.5 


~1 


U 


0.55 


S65 








~2.5 


cc 


1.3 ±0.5 


D17 


2.77 




3.0 


~0.75 


pc 


0.7 


B8 








~1.50 


pc 


1.2 


B8 








~2.25 


pc 


1.6 


B8 


2.76 




14 


~2 


p 


0.91 ±0.06 


P15 








~3 


Al 


2.29 ±0.04 


P15 








—11 


Cu 


2.56 ±0.05 


P15 








12 


PP 


<2.6 


W21 


2.77 




14.5 


~3 


Al 


2.20 ±0.17 


G6 








~11 


Cu 


2.29 ±0.12 


G6 


2.76 



310 



The Neutron 



[Pt. VII 



TABLE 8 (Continued) 
(a) Measured Values of Integral Inelastic Scattering Cross Sections, Oin(E,E t ) 



Ele- 
ment 


E (Mev) 


E t (Mev) 


Detec- 
tor f 


Oin(E,Et) % 

(barns) 


Refer- 
ence 


<r c (barns) 
(Fig. 10) 


Bi 


2.5 


~1 


U 


0.64 


S65 


2.77 




14 


~2 


P 


1.03 ±0.11 


P15 








~3 


Al 


2.28 ±0.08 


P15 








~11 


Cu 


2.56 ±0.05 


P15 








12 


VP 


<3.3 


W21 


2.78 



(b) Temperatures for Observed Distributions of Inelastically Scattered Neutrons 

(Eq. 69) 



Element 


E (Mev) 


Range of 

e (Mev) 


T{E) (Mev) 


Reference 


B 


11 


>4 


2.3±0.3 


G31§ 






1-4 


0.9±0.1 


G31 


Al 


15 


>1 


1.1 ±0.1 


S53 


Si 


10.6 


>2 


1.3±0.1 


G31 


Fe 


1.5 


<0.9 


none || 


B8, F9 




3.0 


<2.25 


none 


B8, F9 




15 


>1 


0.6±0.1 


S53 


Co 


10.5 


>2 


0.95 ±0.1 


G31 


Pd 


14 


>2 


0.85±0.1 


G31 


W 


1.5 


<0.9 


0.35 


B8, F9 




3.0 


<2.25 


0.50 


B8, F9 


Au 


3.0 


<2.25 


(0.33) 


B8, F9 


Hg 


14.6 


>2 


0.8± 0.1 


G31 


Pb 


1.5 


<0.9 


none 


B8, F9 




2.5 


>1 


none 


D17, F9 




3.0 


<2.25 


none 


B8, F9 




4.3 


>1 


none 


M2 



Sec. 2C] 



Interaction with Nuclei 



311 



TABLE 8 {Continued) 

(b) Temperatures for Observed Distributions of Inelastically Scattered Neutrons 

{Eq. 69) 



Element 


E (Mev) 


Range of 

e (Mev) 


T{E) (Mev) 


Reference 




14 


1-3 


—0.8 


W21 




14.3 


>2 


0.78±0.1 


G31 




15 


>1 


0.7±0.1 


S53 


Ki 


4.3 


>1 


none 


M2 




14 


1-5 


—0.9 


W21 



f The symbols have the following meanings : 

7 = direct detection of gamma-rays from the product nucleus 
P = T 31 {n,p) reaction 
Al = Al 27 (n,p) reaction 
Cu = Cu 63 (ra,2w.) reaction 

pc = detection in a biased proton recoil proportional counter 
pp = detection by proton recoils in a photographic emulsion 
cc = detection by proton recoils in a cloud chamber 
U = detection by U 238 fission in an ionization chamber 

t Values in parentheses have not been corrected by the authors (B8) for 
effects of multiple scattering in the target. 

§ Reference (G31) is to measurements of Gugelot on neutron distributions 
from {p,n) reactions. In these cases, the element given is that of the product 
nucleus and the energy (second column) is the maximum neutron energy, i.e., 
the proton kinetic energy plus the reaction Q value. 

|| "None" means that the inelastic scattering cannot be treated by the statis- 
tical theory. Instead, individual levels must be considered. 

behavior of the cross section for the excitation of the metastable level; 
in the second case, the partial cross section for the production of the 
metastable state through the excitation of the ith level is given by the 
product of Eq. (42a) and a factor representing the relative probability 
for the decay of the ith level to the metastable state. The excitation 
function (total cross section vs. neutron energy) is the sum of the partial 
cross sections. It will exhibit discontinuities at the energies correspond- 
ing to the inception of excitation of those product nucleus levels which 
have an appreciable probability of decaying to the metastable level. 

The cross section for the excitation of the metastable state of In 115 
by neutrons of energy up to —4 Mev has been measured by Cohen 



312 The Neutron [Pt. VII 

(CI 7). The resolution of his measurements was too crude to detect the 
effects of individual levels of the product nucleus. More recently Ebel 
(El) has measured the excitation function for In 115 * with good resolu- 
tion. He observes a threshold at 600 kev (no direct excitation of the 
metastable level) and discontinuities corresponding to additional prod- 
uct nucleus levels at 960 and 1370 kev. Ebel has also measured the 
cross section for the excitation of the 540-kev metastable level (half-life 
7 sec) of Au 197 . This level can be directly excited, and also excited 
through levels at 1.14 and 1.44 Mev. 

Since the metastable state is characterized by a large difference in 
spin from the ground state, the levels of the product nucleus which are 
involved in its excitation are, perforce, those whose spins differ appre- 
ciably from that of the ground state. The shape and magnitude of the 
cross section for excitation of the metastable state can be used to deter- 
mine (within limits) the spins and parities of the states involved in its 
excitation (El). 

II. Diffraction or Shadow (Non-Capture) Elastic Scattering 

In the previous discussions, we have considered the total cross section 
as consisting of two parts : (1) that portion which leads to the formation 
of a compound nucleus, <r c , and (2) that portion corresponding to proc- 
esses in which the incident neutron merely changes its direction, with- 
out ever effecting a change in the quantum state of the target nucleus; l 
this elastic scattering, <r e i (see Eq. 68), is variously referred to as diffrac- 
tion or shadow scattering. 

For slow and intermediate neutrons (X > R) capture and diffraction 
scattering are intimately connected because of the interference between 
resonance and potential scattering. For fast neutrons (X < R) the two 
processes separate in a natural fashion. In the first place, the position 
(extent) of the neutron is relatively well defined. Those neutrons which 
strike the nucleus have a high probability of penetrating its surface, 
whereupon they are captured into a compound nucleus. While a cer- 
tain fraction (relatively large for light nuclei, small for medium and 
heavy nuclei) may be re-emitted with the full energy, these (capture) 
elastically scattered neutrons can, except for interference effects in the 

1 The target nucleus, initially at rest, will of course receive momentum and kinetic 
energy as a result of the scattering, and the neutron will, correspondingly, lose energy. 
However, by the term "quantum state" we refer to the state of internal motion of 
the target nucleus, which is unaffected unless the incident neutron penetrates the 
nuclear surface. 




Sec. 2C] Interaction with Nuclei 313 

neighborhood of a resonance, be separated from the diffraction scattered 
neutrons by virtue of their different angular distribution. 

Shadow scattering, on the other hand, results from- the diffraction of 
those neutrons which pass close by, but not into, the nucleus. Thus, 
despite the fact that we are considering the energy range for which 
X < R, in which neutrons may be expected to exhibit a minimum of 
wave properties, it is precisely the wave nature of the incident neutron 
beam which leads to the phenomenon of shadow scattering. Further- 
more, while the angular distribution of the diffraction elastically scat- 
tered neutrons depends on the value of the nuclear radius (specifically, 
on kR = R/X), the nature of the diffraction scattering process is inde- 
pendent of the nuclear atomic number. 

To a good approximation, shadow scattering is analogous to the 
diffraction of a plane wave (say light) by a spherical obstacle, for wave- 
lengths small compared to the size of the obstacle. The scattered neu- 
trons are confined to within a relatively small angle, 

0o « ^ (71) 

An expression for the angular distribution of the scattered neutrons was 
first derived by Placzek and Bethe (PI 7) on the basis of the optical 
analogy. Recently, Hauser and Feshbach (H44) have derived a more 
accurate formula, 

-~^ = ' cot 2 (-){Ji[fc(fl + X) sine]} 2 (72) 

dQ. 4 \2/ 



for the cross section per unit solid angle for scattering of neutrons into 
the angle 9; J t is the Bessel function of the first kind. The cross section 
has a maximum at = 0, falls to zero at k{R + X) sin 6 = 3.83, and then 
goes through a series of subsidiary maxima and minima. 

For very small wavelengths, X <K R, the cross section for shadow 
(elastic) scattering is 



"el 



= fdv(0) « r(B + X) 2 (73) 



At this extreme, a t « 2ir(R + X) 2 (Eq. 67), so that diffraction elastic 
scattering accounts for approximately half of the total cross section for 
fast, very fast, and ultra fast neutrons. 

The experiments of Amaldi, Bocciarelli, Cacciapuoti, and Trabacchi 
(A21) on the angular distribution of the elastic scattering of 14-Mev 



314 

1800 

1700 

1600 

1500 

1400 

1300 

1200 

1100 

1000 

900 

800 

700 

600 

500 

400 

300 

200 

100 



The Neutron 



[Pt. Vii 




v?= 


8.5 
































a(8; 


vs. 6 








_| 






















E-- 


=14 W 


ev 


















































































R='t 


.5 I 


















































^w \ 

to 














































































R = ( 


5.5 












































































R = l 


>.5 


















































R=A 


??~~~ 


































X 


X 


x 




X 




X 


i? = 3.5 









10 15 20 25 30 35 40 45 50 55 60 65 



Fig. 16. Angular distribution in diffraction elastic scattering of 14-Mev neutrons 
for a number of nuclear radii (given in units of 10~ 13 cm). The curves are accord- 
ing to the theory of Hauser and Feshbach (H44). The crosses are from the measure- 
ments of Amaldi and co-workers on lead (A21). 



Sec. 2C] Interaction with Nuclei 315 

neutrons by lead are in excellent agreement with the theory. In addi- 
tion to the general shape of the scattered neutron distribution, they 
have observed subsidiary maxima at approximately the expected angles. 
The theoretical predictions for E = 14 Mev and a number of values of 
the nuclear radius are shown in Fig. 16. Also plotted are the experimen- 
tal points of Amaldi and co-workers (A21). 



III. Angular Distribution of Scattered Neutrons; the 
Transport Cross Section o> 

The angular distribution of the scattered neutrons, with respect to 
the direction of the incident neutrons, is usually described in terms of a 
differential scattering cross section, <r sc (0), the cross section for scatter- 
ing into unit solid angle at the angle 6, in the laboratory system, 



f«c 



= f<Tac(6) da (73a) 



Deviations from spherical symmetry result from a number of possible 
causes, some of which have previously been discussed. Since the angu- 
lar distributions are determined by the nature of the scattering process, 
measurement of o- sc (0) can sometimes lead to useful nuclear information. 
The angular distribution of scattered neutrons is also important in 
determining the rate of diffusion of neutrons through matter (to be dis- 
cussed in a subsequent section). In such problems, the important quan- 
tity is the transport cross section, 

atr = <rt- J <T, C (0) cos 6 da (74) 

a tr determines the rate at which the neutron loses its forward momentum 
(or the memory of its original direction). Under certain special assump- 
tions concerning the nature of the scattering process simple expressions 
for <Tt r can be derived. 

(1) Light Nuclei. In the scattering of fast neutrons by light nuclei, 
elastic scattering is the most important process. Furthermore, the 
scattering is (except at the highest energies) mainly £-wave scattering, 
i.e., spherically symmetrical in the center-of-mass coordinate system 
(c.m.s.). Neglecting all processes but $-wave elastic scattering, the 
angular distribution of the scattered neutrons is still not symmetrical in 
the laboratory system (l.s.), owing to the forward motion of the center 
of mass. 



316 The Neutron [Pt. VII 

Thus, for a neutron, scattered through the angle <p, in the c.m.s., by 
a nucleus of atomic weight A, the angle in the l.s. is given by 



(1 + A cos <p) 

(A 2 + 1 + 24 cos?)' 



cos 6 = , A 2 , , , »„ — VA W 



E (75a) 



The energy in the l.s. of the elastically scattered neutron, of initial 
energy E, is 

r(A 2 + l + 2Acos**r 

E = 

L {A + l) 2 

the difference (E — E') going to the recoil nucleus. The energy of the 
scattered neutron varies between a maximum, E mB ,^' = E, for <p = 8 
= 0, and a minimum, 

_ -(A - iy 2 



Er, 



■'-[ 



(A + 1). 



# b a E (75b) 



for ? = 180°. For <S-wave scattering (in the c.m.s.) the scattered neu- 
trons are distributed uniformly in energy between E and aE. Corre- 
spondingly, under the above assumptions, 

2 

(Ttr = Ot ~ — "el (74a) 

oA. 

(2) Medium and Heavy Nuclei. It is evident from Eq. (74a) that 
the effect of the motion of the c.m.s. can be neglected for medium and 
heavy nuclei. On the other hand, elastic scattering is no longer spheri- 
cally symmetrical, owing to diffraction effects. Most of the capture 
leads to inelastic scattering, and the resulting neutrons are not necessarily 
symmetrically distributed, especially if only a few levels are involved 
(medium nuclei). However, for most purposes, it is reasonable to treat 
the inelastic scattering as though it were spherically symmetrical, and 
to substitute a e i for u sc in Eq. (74). With this assumption the theory 
of Feshbach and Weisskopf (F28) can be used to predict the energy de- 
pendence of atr- The results are shown in Fig. 17, together with the 
experimental values of Barschall et al. (B8). 

(d) Very Fast Neutrons. So far as neutron scattering is concerned, 
the very fast-neutron energy range does not differ in any significant re- 
spect from the fast-neutron range. Most of the preceding discussion 
applies equally well to very fast neutrons; indeed, many of the experi- 
mental results quoted were for neutrons of energy > 10 Mev. However, 
in the extension of the previous considerations into the very fast-neutron 
region, the following points must be kept in mind: (1) Since many levels 
of the target nucleus can be excited by very fast neutrons, except for a 



Sec. 2C] 



Interaction witrfNuclei 



317 



few of the lightest nuclei, the inelastic scattering must be treated accord- 
ing to the statistical theory. (2) Because of the large excitation energy, 
the compound nucleus has, in addition to neutron re-emission, other 
possible modes of decay. Thus, for light and some medium nuclei, 
(n,p), (n,a), etc., reactions compete favorably with neutron re-emission. 
For medium and heavy nuclei, these reactions are also possible, but the 
{n,2ri) reaction is the most probable competing process. Although we 
postpone the detailed discussion of this reaction, it is important to note 














N\^o= 


> 


+ Fe 
o W 
A Pb 




Xo=8 \N 


. +- 








Ir'-^^^l 




■ i 


\ 






A ' : -=:==. 


i i i i 


i i I i 


i i i i 


i i i i 



Fig. 17. a tr vs. x = kR for nuclei of X = KoB = 5 and 8; according to Feshbach 
and Weisskopf (F28). The points, for iron, wolfram, and lead, are from the measure- 
ments of Barschall et al. (B8). 

that its presence has a strong effect on the observed energy distribution 
of the emerging neutrons. Experiments designed to measure the energy 
distribution of inelastically scattered neutrons must, in the very fast- 
neutron range, take into account the deviations from Eq. (69) brought 
about by the presence of the (n,2n) reaction (S53). 

The upper end of the very fast-neutron range is characterized, for all 
nuclei, by X « R. Thus at these energies <x t ~ 2x(fi + X) 2 and c c « 

cr e i » tt{R + X) 2 . 

(e) Ultrafast Neutrons. In the ultrafast-energy region, the character 
of neutron reactions undergoes a gradual change. The simple picture 
of neutron capture into a compound nucleus, as developed in the previ- 
ous sections, is no longer applicable. Instead, as pointed out by Serber 
(S21), the nucleons in the nucleus must be treated as relatively inde- 
pendent particles, since their energy of internal motion and of binding is 
smaller than the energy of the bombarding neutron. Furthermore, 
owing to the rapid decrease of the primary (n-p) and (n-n) cross sections 
with increasing neutron energy — roughly as 1/E in the energy range 50 



318 The Neutron [Pt. VII 

to 250 Mev — the incident neutron has an appreciable probability of 
passing through the nucleus without undergoing any collisions, i.e., the 
nucleus is partially transparent to ultrafast neutrons, the more so the 
lighter the nucleus. 

Under these circumstances the probability of a nuclear reaction can be 
described in terms of an absorption coefficient (reciprocal mean free 
path) of nuclear matter for neutrons, 

3Ao- 

* = ^ Cm (76) 

where 

Za{n-p) + (A - Z)a(n-n) 

a = - (76a) 

A 

[In Eq. (76a) a(n-p) and <r(n-n) are appropriate averages over the spec- 
trum of relative energies of the incident neutron and the nucleons in the 
nucleus, appropriately reduced to take into account the effect, due to 
the exclusion principle, that not all energies are available to the recoiling 
nucleons.] Defining <r„ as the cross section for a collision of the incident 
neutron inside the nucleus, integration over a spherical nucleus of radius 

R yields 

,1 1 - (1 + 2 K R)e- 2KR } 

The nuclear transparency also influences the diffraction elastic scat- 
tering of ultrafast neutrons, both in the magnitude of <?d — <r e i and in 
the angular distribution of the scattered neutrons. These effects have 
been considered in the calculations of Fernbach, Serber, and Taylor 
(F25). They have derived the diffraction scattering of a partially 
transparent sphere, characterized by the propagation vector k + fci 
= [2m(E + V)] 1A /h. (V is the effective nucleon potential inside the 
nucleus; k = 1/X is the incident neutron wave number.) The results 
of these calculations, for the special case kyJK = 1.5, are shown in Fig. 18, 
in which a^, c a , and cr t = a a + <?d are plotted against kR. For large 
values of kR (complete opacity) <r a » ad — > kR 2 . 

Fernbach, Serber, and Taylor have compared their calculated values 
of (T t with the measurements of Cook, McMillan, Peterson, and Sewell 
(C25), for E « 85 Mev. They find excellent agreement, assuming R 
= 1.37A M X 10~ 13 cm, k = 2.2 X 10 12 cm -1 , and k t = 3.3 X 10 12 cm -1 
(V = 30.8 Mev). The measurements of <r t at E « 95 Mev, by DeJuren 
and Knable (D9) are also in good agreement with theory. These investi- 
gators have, in addition, determined a a for two elements (carbon and 
copper), by a "poor geometry" attenuation measurement, and obtained 
values in good agreement with the theory. 



Sec. 2C] 



Interaction with Nuclei 



319 



The angular distributions of 84-Mev neutrons elastically scattered by 
aluminum, copper, and lead have been measured by Bratenahl, Fern- 
bach, Hildebrand, Leith, and Moyer (B59). The results are shown in 
Fig. 19 in which the dotted curves are the predictions of the theory of 
Fernbach, Serber, and Taylor. 

Measurements of the total cross sections of a number of elements have 
been carried out at 280 Mev by Fox, Leith, Wouters, and MacKenzie 
(F41), and at 270 Mev by DeJuren (D10), who also obtained lower 




Fig. 18. Absorption, diffraction, and total cross sections as functions of the nuclear 

radius, measured in mean free paths for the incident neutron. The curves are for 

ki/i< = 1.5, corresponding to neutrons of E « 90 Mev (F25). 

limits for <r of carbon, copper, and lead by "poor geometry" attenuation 
measurements. The above-mentioned measurements of a t are included 
in the compilation of Adair (A2). Cross sections of some representative 
elements have been measured at a number of intermediate energies — 
between 110 and 240 Mev— by DeJuren and Moyer (D12). A rather 
interesting common feature of all the observed <x t vs. E curves is their 
comparative independence on energy above <~160 Mev. Another point 
of interest is the relative constancy of the ratio <r t (270 Mev)/<r* (95 
Mev) ; its value is 0.57 between beryllium and tin and then rises slowly 
to 0.67 for uranium. 

The last feature is difficult to reconcile with the partially transparent 
nucleus model, which predicts considerably greater nuclear opacities 
for the heavier nuclei. 1 Indeed, for the 270-Mev data, it is only possible 

1 This peculiar behavior of <r ( for neutron energies > 200 Mev may possibly be 
associated with the inception of meson production. While the threshold for pion 
creation is ~275 Mev in a nucleon-nucleon collision, meson production is observed 
at considerably lower energies for the bombardment of nuclei by nucleons. This is 
due to the internal motion of the nucleons in the nucleus. 



320 



The Neutron 



[Pt. VII 



o 1 - 





1 


1 ' 
Al 


— r — ' — i — ' I ' I ' i ' 












1 


1, | 


o Carbon detector measurement 




















~^ 


1 








X* 






















1 . 


1,1,1,1 .~-T-^- 



24° 



20 


.... , , 

1 Cu 


r-T — ' "7" i 1 I 1 ' 1 » 


18 




~ 


c 






a is 


1 


" 


o 


,4 




S 14 

to 


- ii c 

"^0 


"" 








a & 


\ 








VI 


1 1° 


- 




- 


\ 
\ 


a 




\ 
\ 


4 




\ 


2 


' 


. I.I, r« J. l-a— 



0° 



12° 16° 20° 



80 

ft 70 

.2 

% 60 

w 

at 

O 50 

O 

1 40 

0) 

S 30 



— i — i — i — i 1 — i — r— i — 

- Pb 


,,.. , _, , 


- 


Ntf 




•- 


\ 
\ 






\ 
\ 

1 i 1 ^-tf-^-e-T— 


T-T---J 


' 



4° 8° 12° 16° 20° 24° 

Neutron scattering angle 



Fig. 19. Differentia] cross sections for elastic scattering of 84-Mev neutrons by 
aluminum, copper, and lead, due to Bratenahl, Fernbach, Hildebrand, Leith, and 
Moyer (B59). The curves are the predictions of the theory of Fernbach, Serber, 

and Taylor (F25). 



Sec. 2C] Interaction with Nuclei 321 

"to obtain a reasonable fit . . . [if] the potential change experienced by 
the bombarding neutron when entering a nucleus [is] dropped to zero" 
(D10). 

2. Radiative Capture (n,-y). The excited compound nucleus, produced 
by neutron capture, can decay to its ground state through the emission 
of one or more gamma-rays. The (n,y) reaction competes with neutron 
re-emission (always possible) and with any other mode of decay which 
is energetically permitted. The relative probability of the (n,y) reac- 
tion is determined by the value of the gamma-ray width, r r , as com- 
pared to the widths for all other possible modes of decay: 

*(n,y) = - 1 (42') 

At the excitations corresponding to neutron binding energies, the 
emission of a gamma-ray takes place in a time of ~10~ 14 sec; corre- 
spondingly (Eq. 27) r T ~ 0.1 ev. This value is relatively independent 
of the atomic weight of the nucleus involved and of the incident neutron 
energy (at least for slow and intermediate neutrons), although con- 
siderable fluctuations, from nucleus to nucleus and from level to level 
in the same compound nucleus, are to be expected. The above lifetime 
refers, of course, only to the emission of the first gamma-ray, after 
which the compound nucleus is usually committed to radiative decay, 
not having sufficient residual energy to decay by any other mode. In 
some cases, successive gamma-ray emission leads to a metastable (iso- 
meric) state of the compound nucleus. Usually, however, the gamma-ray 
emission leads, either directly or through a cascading descent, to the 
ground state of the compound nucleus. 

The study of the energy distribution of the capture gamma-radiation 
is of considerable interest, since it yields information on the level struc- 
ture of the compound nucleus, on the nature of the gamma-ray emission 
process, and on neutron binding energies. Furthermore, knowledge of 
the capture gamma-ray spectrum is of practical importance in the prob- 
lem of shielding neutron chain reactors. However, since further discus- 
sion of these problems at this point would lead us too far afield, we shall 
be content merely to point out that the observation of the capture 
gamma-radiation is one means of detecting the (n,y) reaction. 

In many (n,y) processes the product nucleus is a stable isotope of the 
target, e.g., Cd 113 (n, 7 )Cd 114 ; in such cases, the capture gamma-rays 
are the only reaction products. However, for most nuclei, radiative 
capture leads to an unstable (radioactive) product nucleus. The reac- 



322 The Neutron [Pt. VII 

tion can then be detected and identified by observing the resulting radio- 
activity. Since these radioactive nuclei are most likely to have an excess 
of neutrons, they usually decay by negative beta-ray emission. There 
are a few cases, e.g., Cu 63 (n,7)Cu 64 , in which the product nucleus can 
decay by positron emission or K capture. 

Most of the factors which determine the energy dependence of <r(n,y) 
have previously been described, both in the general discussion of the 
cross section for compound nucleus formation, a c , and in Section 2C1 
on neutron scattering. In the following, we summarize the main fea- 
tures of the radiative capture process. 

(a) Slow Neutrons. The value of <r(n,y) at a given slow-neutron en- 
ergy is determined by two factors: 

(1) The position and characteristics of the closest resonance or, if the 
energy is far from any one resonance (| E — E„ \ ~2> r,-), the positions 
of the closest resonances and their possible interference effects at the 
energy under consideration. The last aspect has been discussed by 
Wigner and co-workers (W29, W30, T5). The behavior of <r{n,y) in the 
vicinity of a resonance has been considered in detail (Section 2B3, espe- 
cially Eqs. 52, 56, 58, and 59). 

(2) The relative value of the gamma-ray width, r 7 , to the total width, 
r = T y + V n + T a + • • • ; in all but a few light nuclei only the first 
two terms, in the above expression for r, differ from zero. Since T y is, 
as per the previous discussion, essentially fixed, the competition between 
radiative capture and neutron scattering is primarily determined by the 
value of r n , which, in turn, is a function of the neutron energy and the 
compound nucleus level spacing, according to Eqs. (37) and (38). 

From these considerations it is clear that the importance and the 
character of the slow neutron (n,y) cross section depend on the atomic 
weight of the target nucleus. In light, medium, and some heavy nuclei, 
levels are widely spaced, T n >?> T y , and radiative capture plays a minor 
role. In most heavy nuclei, on the other hand, levels are closely spaced, 
r 7 2> r„, and radiative capture is the predominant resonance reaction. 
The properties of a number of slow-neutron resonances are summarized 
in Table 9. The values of the resonance energy and level widths are 
given, as well as the deduced values of the level spacings, derived from 
Eq. (37a), for comparison with the observed (average) spacing between 
levels. 

Of particular interest in the study and use (to produce artificially 
radioactive nuclei, for instance) of radiative capture is the so-called 
thermal neutron capture cross section, <r t h.{n,y). This cross section has 
been variously defined, sometimes in a not very precise fashion. We 



Sec. 2C] 



Interaction with Nuclei 



323 



TABLE 9 

Properties of Some Slow-Neutron Resonances — Mainly Capture f 



Target 
Nucleus 


E r (ev) 


I\ (ev) t 


/r„(10- 3 ev)§ 


D* (ev) || 


Dobs (ev) 


17CF 


-75 


0.3 


1.64 X 10 s 


2,200 




26 Mn 8B 


345 


0.2 


12 X 10 3 


7,700 


10,000 


27 Co 59 


123 


0.3 


2.4 X 10 3 


2,800 


10,000 


3oZn 


520 




5-25 X 10 s 


2-8,000 


>500 


3iGa< 69 ' 71 > 


98 


(0.3) 


130 


90 


<100 


32Ge 


95 




3-900 


2-600 




33AS" 


46 


0.05 


110 


120 


50 


35 Br«5,87) 


36 


(0.1) 


35 


86 


30 


42M0 86 


46 


(0.1) 


450 


1,000 


50 


4BRh 103 


1.26 


0.20 


0.45 


6 




4 6 Pd 108 


25 


0.14 


~40 


~60 


>10 


47Ag 107 


15.9 


~0.11 


~5 


~10 


~20 


4rAg™ 


5.17 


0.16 


8.2 


34 


—20 


4 8 Cd 1W 


0.176 


0.115 


0.46 


10 


~25 


49 In 115 


1.45 
3.86 
9 


0.08 
(0.08) 
(0.08) 


1.2 
0.2 
0.5 


14 
2 
5 


6 
6 
6 


61 g b (121, 12 3) 


5.8 
15 


(0.1) 
(0.1) 


~0.3 
~2 


~2 
~8 


7 
7 


B2 Te 123 


2.2 


(0.1) 


8 


70 


~1 


53I 127 


19.4 


0.45 


0.38 


1.2 


15 


6 2 Sm» 9 


0.096 


0.074 


0.31 


13 


2 



324 



The Neutron 



[Pt. VII 



TABLE 9 {Continued) 
Propebties of Some Slow-Neutron Resonances — Mainly Capture f 



Target 
Nucleus 


E r (ev) 


T y (ev) J 


/r„(10- 3 ev)§ 


D* (ev) || 


Dobs (ev) 


63 Eu 161 


-0.011 


0.081 


0.004 


0.5 


~3 


6 3 Eu lB3 


0.47 


0.20 


0.9 


18 


~3 


64 Gd 167 


0.028 


0.12 


0.4 


30 




66 Dy< 161 - 163) 


1.74 


(0.1) 


3 


30 


~3 


7 2 Hf m 


1.08 
2.34 


0.12 
0.16 


0.9 

2.8 


12 

26 


2 
2 


72 Hf 178 


7.6 


(0.1) 


56 


150 




7 3 Ta 181 


6.1 
10.3 
13.6 
20 


0.2 
(0.1) 
(0.1) 


0.0007 
0.85 
0.45 
0.7 


0.04 
3.8 

1.8 
2.2 


5 

5 
5 
5 


74W 182 


4.15 


0.07 


1.1 


4 


~20 


74 W 183 


7.8 


(0.1) 


1.0 


5.3 


20 


74 W(184) 


~200 


(0.1) 


~400 


~200 


~100 


74W 186 


19.25 


0.15 


250 


400 


~150 


76 Re 185 


4.4 
11 


(0.1) 


0.3 
3.5 


2 
15 


5 
5 


76 Re 187 


2.15 
5.9 

7.2 


0.14 


0.43 
0.09 
0.40 


4.2 
0.5 
2.1 


3 
3 
3 


76 0s< 189 > 


6.5 

8.8 


(0.1) 
(0.1) 


1.6 
7.5 


9 
36 


6 
6 


77 I r (Wl. W3) 


0.64 


0.10 


0.2 


4 


2 


77 Ir 193 


1.27 


~0.15 


0.4 


5 


2 



Sec. 2C] 



Interaction with Nuclei 



325 



TABLE 9 (.Continued) 
Properties op Some Slow-Neutron Resonances — Mainly Capture f 



Target 
Nucleus 


E T (ev) 


T y (ev) i 


/r B (io- 3 ev)§ 


D* (ev) |! 


Dobs (ev) 


78 Pt< 196 > 


11.5 
18.2 


(0.1) 
(0.1) 


6.5 
5.5 


27 
19 


10 
10 


79 Au I9r 


4.87 


0.15 


21 


136 


~50 


8oHg (199,201) 


-2.0 
35.5 


(0.1) 
(0.1) 


25 

80 


250 
190 


~15 
~15 


92U 238 


~11 


~0.20 


~8.6 


~20 





t This table represents a complete re-evaluation of the data available in the 
open literature up to the summer of 1952. The measurements, upon which the 
constants are based, are so numerous that we have not attempted to include 
references to them in the table. Instead, the reader is referred to previous com- 
pilations, of which the most complete are those of Blatt and Weisskopf (B34), 
of Wigner (W31), and of Teichmann and Wigner [Phys. Rev., 87, 123 (1952)]. 
Special attention is called to the most recent neutron cross section compilation 
prepared by the AEC Neutron Cross Section Advisory Group and issued by 
the U. S. Department of Commerce, Office of Technical Services, as document 
AECU-2040 (May 15, 1952). We gratefully acknowledge private communica- 
tions of unpublished results by the Harwell time-of-flight velocity selector 
group (M26) and by the Argonne fast chopper group (whose members include 
L. M. Bollinger, R. R. Palmer, and S. P. Harris). Other data, pertaining to 
scattering resonances, can be found in Table 6. 

% The values enclosed in parentheses have been assumed in order to compute 
/T„ from the measured value of <r T 2 . 

§ The numbers quoted are at the resonance energy, except for the negative 
energy resonances, where the neutron width is for the energy E = \E r \. 

|| D* is computed from Eq. (37a): D* = TrK T n /2k. When the statistical 
factor / is not known, it is assumed to be 1 for even-even target nuclei and J 
for odd-A target nuclei. 



326 The Neutron [Pt. VII 

shall define it as the average, over a neutron flux with a Maxwellian 
energy distribution (Eq. 3), of the slow neutron (n,y) cross section, 

,E' 

a(n,y) d4(E,E ) 

<rth(n,y) = ^ (78) 

d<f>(E,E ) 









The cut-off energy, E', is chosen to be sufficiently large so that only a 
very small fraction of the Maxwell distribution is at energies E > E'. 
(Most frequently, the cadmium cut-off, E' ~ 0.3-0.5 ev is used; for E 
= "Fir ev > less than 10 -4 of the neutrons have E > W.) The cut-off is 
introduced for purely practical reasons, since (1) the cross sections above 
~1 ev, which frequently show many resonances, are not well known, 
and (2) most methods for producing thermal neutrons give spectra 
which, above ~1 ev, have a 1/E "tail" superimposed on the Maxwell 
distribution. Unless otherwise specified, thermal neutron cross sections 
are taken to correspond to the Maxwell distribution at a temperature 
of ~300°K (E a = -jV ev, v Q = 2.2 X 10 5 cm/sec). 

If <r(n,y) is a known function of E, a t h can be computed in a straight- 
forward fashion. For nuclei in which the first resonance is relatively far 
from thermal energies, the (n,y) cross section obeys the l/v law, 

(59a) 
and 



(79) 



The thermal neutron (n,y) cross section may deviate from the l/v law 
in a spectacular fashion if a resonance happens to fall in the thermal re- 
gion. The cross section of cadmium, shown in Fig. 20a, is a case in 
point. However, even if the closest resonance does not fall in the ther- 
mal region, it may still cause serious deviations from a l/v behavior for 
thermal neutrons. The cross sections of iridium (Fig. 7d) and of mer- 
cury (Fig. 20b) show, respectively, the effects of a close positive-energy 
(E r > 0) and negative-energy (E r < 0) resonance. Because of such 
effects, considerable care must be taken in interpreting measurements of 
(x t h in terms of tr . 

Table 10 is a collection of thermal neutron (n,y) capture cross sections. 





cr(jl,y) = 

V 






•rofof v 2 e-" y '" ? dv 
•Jo 


^ 


(fth 


r^' 


- (TO 




I v 3 e -v'W dv 


2 



Sec. 2C] 



Interaction with Nuclei 



327 



10,000 

7 
4 



1000 



<= 100 

Is 7 



1 














































Cd 














/ 














































/ 




\ 










Symbol en Reference No. 

Figure in Legend 

• 1 

o 2 


























\ 


































\ 










1 




































\ 


















































\ 
















































\ 
















































V 
















































\ 




































































































\ 
































































o 



































































































































































0.01 



0.1 



1.0 
-E n (ev) 



(1) Rainwater, Havens, Wu, and Dunning, Phys. Rev., 71, 65 (1947). (2) C. T. Hibdon and C. O. 

Muehlhause, Phys. Rev., 76, 100 (1949). Also see W. H. Zinn, Phys. Rev., 71, 575 (1947); Sawyer, 

Wollan, Bernstein, and Peterson, Phys. Rev., 72, 109 (1947). 

(a) 



2 
~ 1 

E ioo 



oo**. 


























I 


n — 


I 






Hg 


P V 


°^oo 






























'°*a 




















1 

2 

♦ 3 
















■\o 


































N 
































^ 








































•v, 










































^ 










































V, 






































'<*■ 














» 


























-VI 






• . • 




. 




















































2 


4 






2 


4 






I 


4 






2 


4 








4 







0.01 



1.0 



100 



•BnCev) 



1000 



(1) W. W. Havens, Jr., and L. J. Rainwater, Phys. Rev., 70, 154 (1946). (2) L. B. Borst el al. (unpub- 
lished). (3) C. T. Hibdon and C. O. Muehlhause, Phys. Rev., 76, 100 (1949). 

(b) 

Fig. 20. Slow-neutron cross sections showing the effects of a resonance at or near 

thermal energies; from Adair (A2). (a) Cadmium, E r = 0.176 ev. Note the sharp 

cut-off at ~0.3-0.5 ev. (b) Mercury, E r = —2.0 ev, showing the influence of a 

close negative-energy resonance. 



328 



The Neutron 



[Pt. VII 



TABLE 10 

Radiative Capture Cboss Sections fob Slow Neutrons 



Element 


At 


<r(n,y) t 
(barns) 


Neutron 
Energy § 


E r for Closest || 

Observed 
Resonance (ev) 


References 1[ 


iH 


1 


0.313 ±0.013 


»0 




R20 




2 


0.00032 


P 




R20 


3L1 


7 


0.033 


th 




W10, H68 


4 Be 


9 


0.0085 


to 


620,000 


R20 


6 B 


11 


<0.05 


th 


430,000 


W10 


6 C 




0.0045 


th 


~4, 000, 000 


R20 




13 


~0.1 


th 




W10 




14 


<200 


th 




W10 


7 N 


15 


0.000024 


th 




tt 


8 




0.00028 ±0.00022 


P 


440,000 


R20 




18 


0.00022 


th 




W10, S22 


9F 


19 


0.0094 ±0.0015 


P 


32,000 


R20, S24 


loNe 




<2.8 


P 




tt 


u Na 


23 


0.50 


th,P 


3,000 


R20, H42, P23, C21 


12M2 




0.057 


P 


230,000 


W10, H42, P23, C21 




24 


0.033 


P 




tt 




25 


0.27 


P 




tt 




26 


0.05 


ih,P 




S24, XX 


isAl 


27 


0.22 


P 


2,300 


W10, H42, P23, C21 


l«Si 




0.13 


P.th 


600,000 


W10, H42, P23, T14, C21 




28 


0.08 


P 




» 




29 


0.27 


P 




tt 




30 


0.12 


th.P 




S24, tt 


15P 


31 


0.17 


P 




W10, P23, C21 


ieS 




0.49 ±0.05 


»o 


111,000 


R20 




34 


0.26 


th 




W10, S24 




36 


0.14 


th 




W10 


17CI 




32 


P 




W10, H42, P23, C21 




35 


42 


th 


-75 


H59 




37 


0.6 


th 


1,800 


W10, S24 


1»A 




0.77 


«o 


>1,000 


R20 




40 


1.2 


P 




W10 


isK 




2.0 


P 


65,000 


W10, H42, C21, P23 




39 


~3 


th 




W10, H31 




41 


1.0 


th 




W10, S24 



Sec. 2C] 



Interaction wtih Nuclei 



329 



TABLE 10 (Continued) 
Radiative Capture Cross Sections for Slow Neutrons 



Element 


At 


(barns) 


Neutron 
Energy § 


E T for Closest [| 

Observed 
Resonance (ev) 


References H 


2oCa 






0.41 


P 


~200,000 


W10, H42, P23, C21 




a 




0.6 


th 




W10, S24 




48 




1 


P 




tt 


2iSc 


45 




12 


P 




W10, P23 






(20 s) 


10 


P 




WW, G17 






(85 d) 


12 


P,th 




W10, S24, ft 


22T1 






5.6 


P 




W10, H42, C21, P23 




46 




0.57 


P 




tt 




47 




1.62 


P 




tt 




48 




7.98 


P 




tt 




49 




1.80 


P 




tt 




50 




0.14 


P.th 




W10, S24, }t 


23V 






4.7 


P 




WW, H42, C21, P23 




51 




4.5 


th 


2,700 


W10, S24 


24O 






2.9 


P 


4,200 


W10, R20, H42, C21, P23 




50 




16.3 


P 




W10, tt 




52 




0.73 


P 




W10, tt 




53 




17.5 


P 




W10, tt 




54 




0.006 


P,th 




WW, S24 


2BMn 


55 




12.6 


P 


345 


W10, H42, C21, P23 


26^6 






2.43 


P 


~10,000 


W10, H42, C21, P23 




54 




2.1 


P 




P22 




56 




3.1 


P 




P22 




57 




0.5 


P 




P22 




58 




0.36 


th 




W10, S24 


27CO 


59 




35 


P 


120 


W10, H42, C21, P23 






(10.7 m) 


0.66 


A 




W10, S24 






(5.3 y) 


22 


th 




W10, S24 


28 Ni 






4.5 


P 


3,600 


W10, H42, C21, P23 




58 




4.23 


P 




P22 




60 




2.70 


P 




P22 




61 




1.8 


P 




W10, tt 




62 




15 


P 




W10, tt 




64 




~2 


ih 




W10, S24 


29C11 






3.6 


P 


~500 


W10, H42, C21, P23 




63 




4.29 


P 




W10, tt 




65 




2.11 


P 




W10, tt 


30Z11 






1.06 


P 


480 


W10, H42, C21, P23 




64 




0.5 


th 




W10, S24 




68 


(13.8h) 


0.9 


a 




W10, H72 






(52 m; 


0.1 


th 




W10, H72 




70 




0.085 


h 




W10, H72 



330 



The Neutron 



[Pt. VII 



TABLE 10 {Continued) 
Radiative Captube Cross Sections fob Slow Neutbons 



Element 



At 



(barns) 



Neutron 
Energy § 



E r for Closest || 

Observed 
Resonance Cev) 



References U 



siGa 



82Ge 



S3A8 
S4Se 



SsBr 



seKr 



3?Eb 



! 8 Sr 



39Y 



71 



70 
72 
73 
74 
76 



(59 s) 
(12 h) 



75 



74 
76 
77 
80 (17 m) 

(59 m) 
82 (67 s) 

(25 m) 



79 (18 m) 
(4.4h) 
81 



78 
80 
82 
83 

84(4.4h) 
(10 y) 



90* 



2.9 
1.4 
3.4 

2.4 
3.3 
0.94 
13.7 
0.60 
0.35 
0.03 
0.2 

4.2 

12.0 

44 

82 

40 
0.5 
0.03 
0.05 
0.004 

6.5 
8.5 
2.9 
2.3 

28 
0.3 
95 
45 
205 
0.1 
0.06 
0.06 
<470 

0.70 
0.7 
0.12 
<200 

1.2 
1.3 
0.005 
<110 

~1 



1.4 



P 

th 
th 

P 

P 

P 

P 

P,th 

P 

P,th 

P,th 



P 
P 
P 
P 

th 
th 
th 
th 

P 

th 
th 
ih 

P 

th 
P 
P 
P 

th 
th 
th 
th 

P 

th 
th 
th 

P 

th 
th 
P 
P 



100-500 



W10, H42, P23 
W10, H72 
W10, S24 

W10, H42, P23 
W10, tt 

» 
U 

tt, S24 
tt 

A33, tt 
S24, ft 

W10, H42, P23, C21, ft 

W10, H42, C21, P23 

tt, S24 

tt, A33 

tt 

W10, S24 

W10, S24 

W10, A33 

W10, A33 

W10, H42, P23, C21 
W10, H72, tt 
W10, H72, ft 
W10, S24 

tt 
W10, tt 

tt 
tt 
tt 

W10, tt 
W10, tt 
W10, tt 
W10, tt 

W10, P23 
W10, S24 
W10, S24 
W10 

W10, H42, P23, C21, tt 

W10, S24 

W10, S24 

W10 

W10 



W10, P23 



Sec. 2C] 



Interaction with Nuclei 



331 



TABLE 10 (Continued) 
Radiative Capture Cross Sections for Slow Neutrons 



Element 


At 


<K»,r) t 
(barns) 


Neutron 
Energy § 


E r for Closest I| 

Observed 
Resonance (ev) 


References H 


«Zr 




0.20 


P.th 




WW, R20, H42, P23, C21 




90 


~0.1 


P 




W10, tt 




91 


1.52 


P 




WW, it 




93 


~0.25 


P 




W10, n 




94 


~0.08 


P 




WW, it 




96 


~0.1 


P 




WW, tt 


4lNb 


93 


1.1 


P 




WW, H42, C21, P23, tt 


42M0 




2.4 


P 




WW, H42, P23, C21, ft 




92 


<0.001 


th 




WW, S24 




95 


13.4 


P 




WW, tt 




96 


1.2 


P 




WW, it 




97 


2.1 


P 




WW, tt 




98 


~0.38 


P.th 




WW, S24, tt 




100 


~0.5 


P.th 




WW, S24, tt 


44R11 




2.5 


P 


9.4 


WW, H42, P23, ft 




96 


0.01 


th 




W10, tt 




102 


1.2 


th 




WW, S24 




104 


0.7 


th 




WW, S24 


«Rh 


103 


150 


P 


1.3 


WW, H42, P23 




(44 a) 


137 


th 




WW, S24 




(4.3 m) 


12 


th 




WW, S24 


46Pd 




8 


P 




WW, H42, P23, C21 




108 


11 


th 


24 


WW, S24 




110 


0.4 


th 




WW, S24 


«Ag 




60 


P,th 




R20, H42, P23, tt 




107 


30 


P 


45 


P22 




109 


84 


P 


5.1 


P22 




(22 8) 


~100 


th 




WW, S24, tt 




(225 d) 


2.3 


th 




WW, S24, tt 


48Cd 




2,400 


»o 




R20, tt 






3,500 


P 




P23 




106 


1 


th 




tt 




110 


0.2 


th 




WW, G17 




113 


19,500 


•0 


0.18 


WW, M39, D22, tt 




114 (2.3 d) 


1.1 


th 




WW, S23 




(43 d) 


0.14 


th 




WW, S23 




116 


1.4 


th 


~100 


WW, S23 


49ln 




190 


H,P 




R20, P23 




113 (72 s) 


2.0 


th 


3.8 


WW, G17 




(50 d) 


56 


th 




WW, S24 




115 (13 s) 


52 


th 


1.44 


WW, S24 




(54 m) 


145 


th 




WW, 824 



332 



The Neutron 



[Pt. VII 



TABLE 10 (.Continued) 
Radiative Capture Cboss Sections for Slow Neutrons 







"(n,y) t 
(barns) 


Neutron 


E r for Closest || 




Element 


a\ 


Energy § 


Observed 


References If 








Resonance (ev) 




soSn 




0.65 


P, »o 




H42, P23, C21, R20 




112 


1.1 


A 




W10, S24 




118 (279 d) 


0.01 


ft 




W10, M29, B55 




120 


0.22 


ft 




W10, S24 




122 (40 m) 


0.30 


ft 




W10, S24 




124 (10 m) 


0.6 


ft 




W10, S24 




(10 d) 


0.15 


ft 




W10, S24 


siSb 




6.4 


P 


5.8 


WIO, H42, P23, C21 




121 


6.8 


ft 




WIO, S24 




123 (1.3 m) 


0.03 


ft 




WIO, D13, tt 




(21m) 


0.03 


ft 




tt 




(60 d) 


2.5 


ft 




W10, S24 


62Te 




4.5 


P 


>300 


W10, H42, P23, tt 




120 


68 


P 




tt 




122 


2.7 


P 




W10, H62, tt 




123 


390 


P 




tt 




124 


6.5 


P 




W10, H62, tt 




125 


1.5 


P 




tt 




126 (9. 3 h) 


0.8 


ft 




W10, S24 




(90 d) 


0.07 


ft 




W10, S24 




128 (72 m) 


0.13 


ft 




W10, S24 




(32 d) 


0.015 


ft 




W10, S24 




130 (25 m) 


0.22 


ft 




W10, S24 




(30 h) 


<0.008 


ft 




W10, S24 


53I 


127 


6.7 


P 


20 


WIO, H42, P23, C21 




129* 


~10 


ft 




W10, tt 




131* 


~600 


ft 




W10, tt 


54X6 




31 


ft 




R20 




128 


<5 


P 




tt 




129 


~45 


P 




tt 




130 


<5 


P 




tt 




131 


120 


P 




tt 




132 


0.2 


ft 




W10, tt 




134 


0.2 


ft 




W10, tt 




135* 


3.5 X 10 6 


P 




U2 




136 


0.15 


ft 




W10, tt 


b&Ca 


133 


29 


P 




W10, H42, P23, R20, tt 




(3h) 


0.016 


th 




W10, S24 




(2.3 y) 


26 


ft 




WIO, S24 




135* 


~15 


ft 




W10, S57 




137* 


<2 


ft 




tt 


6eBa 




1.2 


P 




WIO, H42, C21, P23 




130 


~3 


ft 




WIO, K3, tt 




132 (>20 y) 


~6 


ft 




WIO, K3, tt 




138 


0.5 


ft 




WIO, S24 




139* 


4 


Ik 


W10, Yl, ft 



Sec. 2C] 



Interaction with Nuclei 



333 



TABLE 10 {Continued) 
Radiative Captuke Cross Sections for Slow Neutrons 



Element 


A\ 


<r(.n,y) t 
(barns) 


Neutron 
Energy § 


E r for Closest || 

Observed 
Resonance (ev) 


References U 


srLa 




8.9 


P 




W10, H42, P23 




139 


8.4 


th 




WIO, S24 




140 « 


~3 


th 




W10, K3 


csCe 




0.8 


P 




WIO, H42, P23 




138 


~0.4 


th 




tt 




140 


0.27 


th 




W10, K3, tt 




142 


0.85 


th 




W10, K3, tt 


69Pr 


141 


11.2 


P 


(~10?) 


W10, P23 


eoNd 




44 


P 




W10, H42, P23, tt 




142 


<12 


P 




W10, H55 




143 


240 


P 




W10, H55 




144 


<15 


P 




W10, H55 




145 


<30 


P 




W10. H55 




146 


1.8 


th 




W10, B54, tt 




148 


3.7 


th 




W10, B54, ft 




150 


<45 


P 




W10, H55 


eiPm 


147* 


~60 


th 




W10, P4 


B2Sm 




6,500 


to 




R20 






10,000 


P 




W10, H42, P23 




149 


~50,000 


P 


0.096 


W10, tt 




151 


~7,000 


P 




tt 




152 


150 


th 


10 


W10, S24, B54, tt 




154 


5.5 


th 




W10, S24 


68Eu 




4,200 


P 




W10, P23 




151 


~9,000 


P 


-0.011 


W10, H50, tt 




(9 h) 


1,400 


th 




W10, S24 




152* 


5,500 


P 




W10, H50, tt 




153 


420 


P 


0.465 


W10, H50, tt 




154* 


1,500 


P 




WIO, H50, tt 




155* 


14,000 


P 




W10, H50, tt 


64Gd 




36,000 


P 


0.03 


WIO, H42, C21, P23 




152 


<125 


a 




WIO, S24 




155 


70,000 


p 




WIO, L4 




157 


120,000 


p 




WIO, L4, tt 




158 


~4 


th 




WIO, B77, tt 




180 


~0.15 


th 




WIO, B77, tt 


« S Tb 


159 


44 


P 




WIO, P23 


eeDy 




890 


P 


-1.01;1.74 


P23 






1,150 


»o 




WIO, B65 




164 (1.3 m) 


3,000 


th 




WIO, tt 




(2.4h) 


2,600 


th 




WIO, S24, tt 




165* 


5,000 


th 




WIO, K10 



TABLE 10 (Continued) 
Radiative Captuee Ceoss Sections foe Slow Neutrons 



Element 


A\ 


<r(n,-y) t 
(barns) 


Neutron 
Energy § 


E r for Closest || 

Observed 
Resonance (ev) 


References IF 


!7Ho 


165 


64 


P 




WlOy P23 


esEr 




166 


P 


~0.5 


W10, P23 




170 


>7 


th 




W10, B54 


6»Tm 


169 


118 


P 




W10, P23 


voYb 




36 


P 




W10, P23 




168 


30,000 


th 




W10, A35, tt 




174 


60 


th 




W10, A35, tt 




176 


6.5 


th 




W10, A35, tt 


7lLu 




108 


P 




W10, P23 




175 (3.7 h) 


25 


th 




W10, S24, tt 




176 


4,000 


th 




W10, S24, ft 


72Hf 




120 


P 


~1.0 


W10, H42, P23, C21 




177 


500 


P 




tt 




180 


10 


th 




W10, S24 


7sTa 


181 


21 


P 


4.1 


W10, H42, P23, C21 




(16 m) 


0.030 


th 




W10, S24, tt 




(117 d) 


21 


th 




W10, S24 


7»W 




19 


P 




W10, H42, P23, C21 




180 


~2 


th 




L19 




182 






4 






183 






7.4 






184 


2.1 


th 




W10, S24 




186 


40 


th 


19.5 


W10, S24, tt 




187* 


~80 


th 




L19 


76Re 




84 


P 


2.3 


W10, P23, tt 




185 


100 


th 




tt 




187 


75 


th 




tt 


760s 




14.7 


P 


6.5 


W10, P23 




184 


~20 


th 




L19 




190 


8 


th 




W10, S24 




192 


1.6 


th 




W10, S24 




193* 


~190 


th 




L19 


77lr 




440 


P 


0.64 


W10, H42, P23 




191 (1.5 m) 


260 


th 




W10, G15, tt 




(70 d) 


740 


th 




W10, S24, tt 




193 


130 


th 


(1.3) 


W10, S24 


7 8 Pt 




8.1 


P 


11.5 


W10, H42, P23, C21, tt 




192 


90 


th 




W10, S24, tt 




196 (18 h) 


1.1 


th 




W10, S24 




(82 d) 


0.055 


th 




tt 




198 


3.9 


th 




W10, S24 



334 



TABLE 10 (Continued) 
Radiative Capture Cross Sections for Slow Nbutbons 



Element 


At 


»(n,7) t 
(barns) 


Neutron 
Energy § 


E, for Closest |1 

Observed 
Resonance (ev) 


References *[ 


79A11 


197 


95 


P.th 


4.8 


W10, H42, P23, S24, tt 




198* 


~16,000 


th 




ft 


soHg 




400 


«o 




G20 






340 


P 


-2.0 


W10, H42, P23, tt 




196 


3,100 


P 




W10, 11 




199 


2,500 


P 




W10, 11 




200 


<60 


P 




W10, 11 




201 


<80 


P 




W10, 11 




202 


3.0 


th 




W10, S24, tt 




204 


0.43 


th 




W10, S24, tt 


siTl 




3.3 


P 


260 


W10, H42, P23 




203 


8 


th 




W10, S24, tt 




205 


0.10 


th 




W10, S24, tt 


82Pb 


206 


0.17 


P 


130,000 


W10, H42, P23, C21, ft 




208 


0.0006 


th 


350,000 


W10, tt 


83B1 


209 


0.032 


P 




W10, H42, tt 




(5d) 


0.017 


th 




W10, S24, tt 


joTh 


232* 


7.0 


P.th 




R20, tt 


i 2 U 




3.5 


TO 




TJ2 




235* 


101 


no 




D2 




238* 


2.80 


m 


11 


U2 


94PU 


239 * 


361 


n 




V2 


96Am 


241* 


890 


P 




H29 




(16 h) 


570 


P 




H29 



t The atomic number A refers to the target nucleus. When more than one activity results from the radiative capture 
the half-life of the particular activity, to which the figures in that row pertain, is shown in parentheses. An asterisk 
indicates that the target nucleus is itself radioactive. 

% Unless otherwise indicated, the uncertainty can be assumed to be in the last significant figure. 

§ The symbols have the following meanings: 

to = neutrons of velocity 2200 m/sec (E = -£§ ev) 

P = pile neutrons, usually indicating measurement by the "danger coefficient" method 

th = thermal neutron distribution for a temperature of ~300°K (usually indicates measurement by the "activation" 
technique) 

lj The closest observed resonance is not necessarily the closest resonance, since most elements have not been carefully 
investigated above ~I0 ev. Negative resonances are shown when known. 

H This compilation leans heavily on the excellent and complete collection of nuclear data by Way and co-workers 
(W10). The compilation of Ross and Story (R20), although unfortunately somewhat out of date, has been most useful. 

ft This table has been checked against a preliminary version of the extensive collection of the A.E.C. Cross Sections 
Committee, D. J. Hughes, Chairman. A number of values from that table have been added to this one. Furthermore, 
we have inclined somewhat to the choices, between alternative values, made by that committee. The responsibility 
for the choices, however, rests on our shoulders. We gratefully acknowledge our indebtedness to D. J. Hughes and 
his committee. 

Jt We are grateful to H. Pomerance of the Oak Ridge National Laboratory for making available unpublished measure 
ments of his group based on the "danger coefficient" method (P23) 

335 



336 



The Neutron 



[Pt. VII 



(b) Intermediate Neutrons. For light and medium nuclei, (n,y) cross 
sections for intermediate neutrons are very small, since T y « r„. In 
heavy nuclei, the radiative capture cross sections are still quite appre- 
ciable in this energy range. The average (over many resonances) cross 
section is given by Eqs. (60), (62), (62a), and (63) : 



_ 2w^T n V y /600\ /iy\ 



500\/iy 

II- — I barns 



(60a) 



(E is in ev). Thus, the cross section follows a 1/v law at low energies, 
and a 1/E law at high energies, the transition occurring when T n « T y ; 
for heavy nuclei, this occurs in the intermediate region. 

















i 






































































8 




















































































-I- 


























-« 
























































6 
5 














































.(T t J 




-•- 


































4 












































































-d 


























































































3 


























































































































































































































































t 




















































































»1.0 

H 

St 8 


























® 














































































































































































































































































































■" r 




















































































-, b 














































f 


















































































1 






































4 












































a 












































ol on Reference 


No." 
d 


























































3 




Sj 


111U 
































































Figure in Leger 
• 1 




























































2 
































































x l l 
































































© 4 
o 5 




























































































































8 






























































































































6 
5 





































































































































































































































4 






































































































































































3 





































































































































































0.1 
^7„(Mev) - 



10 



(1) R. Fields e( at., Phys. Rev., 71, 508 (1947). (2) H. Aoki, Proc. Phys.-Math. Soc. Japan, 21, 232 

(1939). (3) J. R. Dunning et al., Phys. Rev., 48, 265 (1935). (4) E. Segre et erf. (unpublished). (5) 

J. Marshall and L. Szilard (unpublished). 



Fig. 21. Radiative capture cross section, a-(n,y), for iodine in the intermediate- 
neutron energy range. Also shown is the total cross section, 07. From Goldsmith, 

Ibser, and Feld (G20). 

(n,y) cross sections for intermediate neutrons have not been so exten- 
sively investigated as those for slow neutrons. Among the light and 
medium elements, fluorine, aluminum, and vanadium have been studied 
by Barschall and co-workers (see the compilation of Adair, A2); the 
<r(n,y) vs. E curves show the expected resonances; the magnitude of 



Sec. 2C] Interaction with Nuclei 337 



<j(n,y) is small — of the order of millibarns. Curves of a{n,y) vs. E have 
been obtained, for a number of heavy elements, by Segre and co-workers; 
their results are included in the compilation of Goldsmith, Ibser, and 
Feld (G20). One of these, for iodine, is reproduced in Fig. 21. The cross 
section shows the expected behavior (see Eq. 60a). Measurements on 
fifteen isotopes, using Sb-Be neutrons (E ~ 35 kev), have been reported 
by Hummel and Hamermesh (H77). 

(c) Fast Neutrons. In the fast-neutron energy region, the behavior of 
<r(n,y) will differ from that described in the previous section for two 
main reasons: (1) With the inception of the possibility of inelastic scat- 
tering, the competition in the de-excitation of the compound nucleus 
becomes even less favorable to the radiative process. (2) For fast neu- 
trons the possibility of compound nucleus formation by neutrons of 
I > becomes appreciable, so that relationships such as Eq. (62b), 
based upon I = capture only, are no longer strictly valid. This factor 
tends to compensate for the decrease of radiative capture due to inelas- 
tic scattering, since the inelastic re-emission of I > neutrons is ener- 
getically unfavored. 

Hughes, Spatz, and Goldstein (H72) have made a systematic survey, 
covering 32 isotopes, of (n,y) cross sections at an energy of ~1 Mev. 
Their results are summarized in Fig. 22. The most significant features 
of these results are: (1) a rapid increase of c{n,y) with A (roughly expo- 
nential), from «1 millibarn at A « 35 to =200 mb at A « 110; (2) 
roughly constant (n,y) cross sections, of ~100 mb, for A > 120; (3) 
marked deviations from the norm for target nuclei containing neutron 
closed shells (e.g., Ba 138 , 82 neutrons; Pb 208 and Bi 209 , 126 neutrons). 
These magic number nuclei have anomalously small (n,y) cross sections 
of ~2-3 mb. Since the radiative capture cross sections at ~1 Mev are 
essentially inversely proportional to the compound nucleus level spacing 
at the excitation energy (e + 1) Mev (see Eqs. 60a and 37, with T 
~ T n » T 7 ), the general dependence on atomic number is in reasonable 
agreement with expectation. The anomalous behavior of the magic 
number nuclei reflects their small binding energies for an additional 
neutron and, possibly, large level spacing. 

Hughes et al. (H72) have also surveyed the available data on the de- 
pendence of a(n,y) upon neutron energy, for energies between ~0.1 and 
~10 Mev. While the results of the different investigators are in rather 
poor agreement with regard to the absolute values of a(n,y), the relative 
values seem to follow, roughly, a ~l/i? law. There are, however, a 
number of unexplained exceptions, notably In 115 , in which o(n,y) ap- 
pears to be constant over the energy range 0.1 to 1 Mev. 



338 



The Neutron 



[Pt. VII 



There are almost no data available on radiative capture in the very 
fast- and ultrafast-neutron regions. At these energies, a{n,y) is expected 
to be very small for all nuclei. 



20 40 



100 120 140 160 180 200 220 



1.0 - 



0.1 



1 1 ' 1 


i i 




1 ■■T 


— 1 1 1 

OLu 




o 
OAg 


n 




S & 










™ O An 


- 


Kl. . 


-oi 




.. ° f 


/^Ag 


°Sb 




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• 


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Br / , 




ONci 




O / Nb 
/ •* 






• 




V OK u 




ONd 




G^f.*^ 




* 






OBf • 








Gu/ 
Co°°/ 


1 Mo 
9Kr°M<, 




OPr 




/S Cu 

/ °Ni 


OR u 


LaS^ 




k>Mn 


* 




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MgO/ 










PA\ 

i i i 


1 1 




1 1 


, 1 , 



40 60 



100 120 140 160 
A 



180 200 220 



Fig. 22. Activation (n,y) cross sections for fission neutrons (~1 Mev, average 
energy) vs. atomic weight, A. The points lying appreciably below the smooth curve 
all correspond to neutron numbers near or at one of the "magic" values — 50, 82, 126. 
We are indebted to Hughes, Garth, and Eggler (private communication) for this 
figure. Earlier results are reported and discussed by Hughes, Spatz, and Gold- 
stein (H72). 

3. Charged Particle Reactions. After the capture of a neutron, the 
compound nucleus can sometimes decay by emission of a charged parti- 
cle. Among the possible reactions, (n,p) and (n,a) are most frequently 
encountered. The energy dependence of a charged particle reaction is 
governed by Eq. (42) : a(n,a) = <r c r o /r. 

(a) Slow Neutrons. For a charged particle reaction to take place with 
slow neutrons, it is necessary that the reaction be exoergic (Q > 0). 
Furthermore, if the reaction is to compete favorably, the available 
energy must be sufficiently great to allow appreciable penetration of 
the Coulomb barrier (the factor G a in Eqs. 32 and 37). These considera- 
tions limit the observable slow-neutron (n,p) and (n,a) reactions to light 
nuclei. The properties of the most important slow-neutron charged 



Sec. 2C] 



Interaction with Nuclei 



339 



particle reactions are summarized in Table 11. The cross sections in 
the third column refer to the isotopes involved in the reaction; these 
must be multiplied by the relative abundances (fifth column) to obtain 
the cross sections of the normal elements. 

TABLE 11 

Properties of Exoergic (n,p) and (n,a) Reactions 



Reaction 


Q Value 
(Mev) 


Isotopic Cross 

Section at 
v = 2.2 X 10 B 
cm/sec (barns) 


References 


Relative 

Abundance of 

Isotope in 

Normal 

Element (%) 


He 3 (»,p)H 3 

Li 6 (n,a)H 3 

B w (n,a)Li 7 

N 14 (n,p)C 14 
Cl 35 (n,p)S 36 


0.7637 

4.785 

2.791 

0.626 

0.62 


5060 ± 200 

910 ± 100 

3770 ± 110 t 

1.76 ±0.05 

~0.3 


T16, C26, K14 
T16, R20 
T16, R20 
T16, C26 
W10 


1 - 10 X 10" 5 
7.4 
18.83 
100 

75.4 



f A more recent value of the B 10 (n,a) cross section is 3990 b (AEC Neutron 
Cross Section Advisory Group, AECU-2040, U. S. Department of Commerce, 
May 15, 1952). This cross section is of special significance since a majority of 
the quoted thermal neutron absorption cross sections, in this and in Table 10, 
are based on a comparison with boron absorption. Thus a change in the ac- 
cepted value of this cross section is directly reflected in a change, of equal frac- 
tional magnitude, in many of the other values quoted. 

The cross sections are given at a single neutron energy, 0.025 ev. 
Since these reactions fully satisfy the conditions for Eqs. (58) and (59), 
they can be assumed to follow the 1/v law in the slow-neutron range. 
Owing to the large level spacings of such light nuclei, the first resonances 
occur well into the intermediate- or even the fast-neutron region. Fur- 
thermore, owing to the large Q values (available charged particle energy) 
of these reactions, the reaction widths are essentially constant over a 
wide energy range. Thus, these reactions obey the 1/v law over a com- 
paratively broad energy region which, for the B 10 (n,a) reaction, for in- 
stance, extends to > 10 4 ev. 

(b) Intermediate Neutrons. For intermediate neutrons, charged parti- 
cle reaction cross sections depart from the 1/v law because (1) resonances 
are present in or close to the intermediate-energy region, and (2) the 
particle width, r , is no longer independent of the neutron energy. A 
resonance in the cross section for the formation of the compound nucleus 
is, of course, also a resonance in the reactions involved in the compound 



340 



The Neutron 



[Pt. VII 



nucleus decay, including the charged particle reactions (see, for in- 
stance, the compilation of Adair, A2, figures 8, 9, and 12). 

An interesting example of charged particle reaction resonances is N 14 , 
which has been investigated with good resolution for neutron energies 
between ~0.2 and 2 Mev, and is shown in Fig. 23. In addition to the 
(n,p) reaction, previously discussed, the N 14 (n,a)B u reaction is also in 
evidence. This reaction is slightly endoergic (Q= —0.26 Mev), and 
does not have an appreciable cross section below ~1 Mev. 



One of the most striking features of Fig. 23 is the apparent separation 
of the resonances (corresponding to the decay of the same compound nu- 
cleus) into predominantly (n,p) — e.g., 1.4 Mev— and (n,a) — 1.8 Mev — 
resonances. Although at first glance this may appear to be in contra- 
diction to the ideas of the compound nucleus picture — upon which we 
have leaned so heavily — the observations are, as will be seen from the 
following discussion, consistent with our present notions, if proper 
account is taken of the angular momentum and parity properties of the 
nuclear levels involved. 1 

The nuclei involved are N 14 , C 14 , B 11 (all in their ground states), and 
N 15 (in various excited states). The spins of the nuclei are: N 14 , 1=1; 
C 14 , I = 0; B 11 , I = f; and their ground-state parities are: N 14 , prob- 
ably even (assumed 3 D, from the magnetic moment and to explain the 
long half-life for the C 14 beta-decay) ; C 14 , even; B 11 , probably odd (from 
the magnetic moment). Assuming these parity assignments, the parity 
of the level of N 1S , involved in the resonance, completely determines the 
I value of the captured neutron, even I values being associated with the 
states of even parity, and odd I values with odd states. From the laws 
of conservation of parity and angular momentum, the lowest possible I 
value of the emitted proton or alpha-particle is uniquely determined 



according to the following scheme 
involved) : 

V 1 3 

7 \ 2 2 



(J is the spin of the N 15 state 



7 






2 .. .. l p 

■i- 1 • • * • la 


1 


1 1 3 .. l p 

2 2 .. l a 


2 


2 2 4 l p 

1 1 1 3 L 



1 The author is indebted to Professor J. M. Blatt, who first called to his attention 
this possibility for explaining the N 14 resonance separation. A similar discussion 
has been presented by Johnson and Barschall (J5). 



Sec. 2C] 



Interaction with Nuclei 



341 




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342 The Neutron [Pt. VII 

It is evident that for low-neutron energies and, correspondingly, low 
energies of the emitted proton and (especially) alpha-particle (so that 
small I values are favored for both incoming and outgoing particle) the 
resonances divide into two groups: those for which J = \, favoring pro- 
ton emission; levels with J > § , which favor alpha-emission. 

The same sort of arguments can be carried through for different 
assumptions concerning the parities of the nuclei. In particular, the 
assumption of odd parity for the N 14 ground state (other states same 
as above) leads to a reversal in the division between proton-favored and 
alpha-favored levels; i.e., the absorption of a thermal neutron {l n = 0) 
is followed by the emission of an I = 1 proton, etc. Unfortunately, the 
available data do not permit a choice between the two possibilities for 
the parity of N 14 (J5, J7). 

(c) Fast Neutrons. As the energy available to the charged particle 
becomes greater, the Gamow barrier penetration factor approaches 1, 
and charged particle emission is less inhibited. Thus, reactions which, 
although exoergic or only slightly endoergic, have very small cross sec- 
tions for slow and intermediate neutrons become appreciable in the 
fast-neutron region. The N 14 (n,a) reaction, discussed above, is one 
such case. Another example is Ne 20 (w,a)O 17 (J6). 

There are a number of endoergic charged particle reactions whose 
thresholds, E t = — Q(A + 1)/A, fall in the fast-neutron energy range. 
The cross sections for these reactions have a characteristic energy de- 
pendence, rising rapidly (from o- = at the threshold) to a more-or-less 
constant value for energies greater than the "height" of the Gamow 
barrier. Figure 24 shows the measured (»,p) cross sections of two reac- 
tions (on P 31 and S 32 ) whose thresholds fall at ~1 Mev. 

The energy dependence of such "threshold reactions" is, at least for 
energies below the barrier heights, primarily determined by the probabil- 
ity for penetration of the Coulomb barrier by the emerging charged 
particle. The barrier penetration factor is a monotonically increasing 
function of the available energy (E — E t ) and, for a given value of E, 
strongly dependent on the angular momentum of the charged particle; 
the smaller the angular momentum, the greater the penetration factor. 
The energy dependence of the penetration factor has been discussed by 
a number of authors (B24, B34, B40), and by Morrison in Part VI. In 
general, if the reaction can proceed with the emission of charged parti- 
cles of zero angular momentum, it will prefer to do so, and the energy 
dependence near threshold will be determined by Go, the Gamow factor 
for I = particles. In this case the barrier height is 

zZe 2 
Bo = — (80) 



Sec. 2C] 



Interaction with Nuclei 



343 



where z and Z are the atomic numbers of the outgoing particle and 
product nucleus, respectively, and the nuclear radius is given by (B34) : 



and 



R = 1.474* X 10~ 13 cm for protons 

R = (1.304* + 1.2) X 1(T 13 cm for alpha-particles (80a) 



Table 12 lists the properties of a number of useful (n,p) and (n,a) thresh- 
old reactions computed on the assumption of I = outgoing particles. 

TABLE 12 

Properties or Fast-Neutron Threshold Reactions 

Based on computations by Feld, Scalettar, and Szilard, (F8), and Kiehn, (Kll) 





Product 




E (Mev) 


E (Mev) 




Reaction 


Nucleus 


E t (Mev) 


for 


for 


E t + £o 




Half-Life 




Go = 0.1 


Go = 0.5 


(Mev) 


P 31 (n,p)Si 31 


2.7h 


0.97 


2.8 


3.8 


5.3 


S 32 (n,p)P 32 


14.3 d 


0.96 


3.0 


4.1 


5.6 


AF(w,p)Mg 27 


10 m 


1.96 


3.5 


4.5 


5.9 


Si 28 (n,p)Al 28 


2.3m 


2.7 


4.4 


5.4 


6.9 


Fe 66 (w,p)Mn 56 


2.6h 


3.0 


6.3 


7.6 


9.4 


P 31 (»,a)Al 28 


2.3m 


0.91 


6.6 


8.3 


9.8 


AF(n,a)Na 24 


14.9 h 


2.44 


7.5 


9.1 


10.9 



The actual energy dependence of a given charged particle reaction is 
determined, in addition to the barrier penetration factor (in T a ), by the 
cross section for the formation of the compound nucleus, <r c . Among 
the important aspects of the dependence of <r c on E, the presence of 
resonances — especially in the light and medium nuclei, with which we 
are concerned — will be reflected in the reaction cross section; some of 
the resonances in a c may appear weakly or not at all in the reaction 
cross section because of the angular momentum and parity properties 
of the levels involved. 

Although the Gamow penetration factor for the outgoing charged 
particle becomes relatively constant (it slowly approaches one but is 
prevented from being strictly constant or equal to one by the increasing 
importance of higher angular momenta and their associated angular 
momentum barriers) after the available energy exceeds the barrier 
height, B , the reaction cross section will not remain constant as the 
neutron energy is indefinitely increased. In the very fast- and ultrafast- 



344 



The Neutron 



[Pt. VII 



























8 














































6 






























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ff t 
































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(1) R. Fields ef a*., Phys. Rev., 71, 508 (1947). (2) H. Aoki, Proc. Phys.-Math. Soc. Japan, 21, 232 
(1939). (3) E. D. Klema and A. O. Hanson, Phys. Rev., 73, 106 (1948). (4) R. F. Taschek (unpub- 
lished). (5) E. Bretscher et al. (unpublished). 

Fig. 24. The (n,p) cross sections of P 31 and S 32 in the fast-neutron region. Meas- 
urements between 1.4 and ~6 Mev, with poor energy resolution; these curves illus- 
trate the "threshold" behavior of endoergic charged particle reactions. Curves 
from Goldsmith, Ibser, and Feld (G20). 



1 1 a (n,p) 



P SI (n,p)Si SI 




1.5 

> ^ a (n,p) 
30-l(T 26 cm 2 T- 



S S2 (n,p)P s 



3.5 MeV 




3.0 3.5 MeV 

Fig. 24 (Continued). Good resolution measurements between ~2 and 3.5 Mev, 
showing the effect of resonances in the compound nucleus. Curve for P 31 due to 
Ricamo (R9); S 32 curve due to Liischer, Ricamo, Scherrer, and Zunti (L25). 

345 



346 The Neutron [Pt. VII 

neutron energy regions additional reactions, involving multiple neu- 
tron as well as charged particle emission, become energetically possible, 
and the cross section for any given charged particle reaction will, in 
the ensuing competition, decrease with increasing neutron energy. 

4. Fission. For nuclei of A > 130 the binding energy per nucleon de- 
creases gradually with increasing A (Section 2B1). Consequently, most 
of the heaviest nuclei are energetically unstable against division into two 
fragments, i.e., M(A) > M(A — a) + M(a), provided that one of the 
fragments is a relatively tightly bound particle such as an alpha-particle 
or O 16 nucleus. However, as is well known, spontaneous nuclear disinte- 
grations are only (with two exceptions) observed in the heaviest, the 
naturally radioactive, elements. (Many nuclei, lighter than lead, which 
disintegrate by alpha-particle emission, have been produced artificially; 
but their lifetimes are much too short for them to be found in nature, 
even if they had been present in the original distribution of the elements.) 

It is, of course, the Gamow barrier against charged particle emission 
which impedes the spontaneous disintegration of the heavy elements, 
with the effect that alpha-emission is a relatively long-lived process and 
the emission of heavier fragments proceeds at an unobservably slow rate. 

There is, however, one mode of disintegration which involves the 
release of such a large amount of energy that the disintegration can, for 
the heaviest nuclei, proceed over the top of the barrier. This process is 
fission, or the division of the nucleus into two approximately equal 
fragments. 

Consider a heavy nucleus, say Z = 92, A = '240. The binding energy 
per nucleon is =7.5 Mev. Division into two fragments, e.g., Z t = Z 2 
= 46 and A 1 = A 2 = 120, produces two nuclei which are approximately 
at the peak of the binding energy curve with, however, a considerable 
neutron excess, so that their average binding energy per nucleon is some- 
what less than that of the stable nuclei in this region, say = 8.3 Mev. 
The energy release is then ~240 X 0.8 ~ 200 Mev. The barrier height 
against separation of the fragments is 



ZiZ%e Z\Z 2 

R^+R 2 = A x v * + A 2 A 



B = „ = ^ u ( 2mc2 ) ~ 20 ° M ev (80' ) 



[We have used the approximation R = (e 2 /2mc 2 )A H .] However, be- 
cause of the large charge of the fragments, the barrier is very "wide," 
and the lifetime against fission is a very strong function of the differ- 
ence between the barrier height and the available energy, being ex- 
ceedingly long for barriers only a few (~5-10) Mev higher than the 
reaction energy, and exceedingly short for energies above the barrier. 



Sec. 2C] Interaction with Nuclei 347 

The greater the atomic number of the nucleus, the smaller is the margin 
of barrier height over reaction energy. This increase of instability with 
atomic number places an upper limit on the possible Z of the heaviest 
nuclei which can be found among the naturally occurring elements; 
nuclei with Z > 100 would be unstable against spontaneous fission 
(B46). 1 

For the heaviest of the naturally occurring nuclei, then, stability 
against fission depends on the slight deficiency in the available energy 
as compared to the barrier height. The addition of only a relatively 
small energy can serve to push the reaction over the barrier. The ab- 
sorption of a neutron is one means of supplying energy, since capture is 
accompanied by release of the neutron's binding energy. If the binding 
energy is sufficiently great, as in the addition of a neutron to some odd- 
neutron nuclei, fission can be induced by the capture of a thermal neu- 
tron. Among the nuclei which undergo fission after thermal neutron 
capture are IT 233 , U 235 , Pu 239 (S42), Am 241 (C35), and Am 242 (H30). 

Since the energy release is so great, the fission widths, T f , for such 
nuclei should be essentially independent of the neutron energy. Thus, 
in the absence of close resonances, the thermal neutron fission cross 
section should follow a 1/v law. However, for such heavy nuclei the 
compound nucleus level spacing is expected to be relatively small. For 
U 235 , a t >,(n ,fiss.) = 545 barns (U2); additional data are given in Sec- 
tion 4D2. 

In most of the heaviest nuclei, however, the energy release accompany- 
ing neutron capture is insufficient to cause fission. For such nuclei, 
fission can be induced only if the neutron also carries with it a certain 
amount of kinetic energy, so that E + e + Q > B . The fission cross 
sections of these nuclei exhibit comparatively sharp thresholds. Many 
of the thresholds are in the fast-neutron region, and the corresponding 
nuclei make excellent "threshold detectors." Some of these nuclei, and 
their fission thresholds, are collected; in Table 13. Curves of o- fes . vs. E 
for U 238 and Np 237 are shown in Fig. 25. 

TABLE 13 

Approximate Values of Fast Fission Thresholds 





Threshold 




Threshold 


Nucleus 


(Mev) 


Nucleus 


(Mev) 


Bi 209 


60 


TJ238 


1.1 


Th 232 


1.3 


Np 237 


0.4 


Pa 232 


0.5 







1 The experimental data on spontaneous fission are summarized by Segre [Phys. 
Rev., 86, 21 (1952)]. 



348 



The Neutron 



[Pt. VII 



1.5 


Np 23 


7 
















































































• 






































" m 








• 


















































V 
































































*f 




























































































































































































































































































































































































































0.5 



















































































































































































































































































































































































0.5 



1.0 



1.5 
E (Mev) 

(a) 



2.0 



0.7 
0.6 



SO-6 



8 0.3 



tj2S8 









































































































































































0.2 0.3 0.4 0.6 0.8 1.0 2.0 

Neutron energy (Mev) 

(b) 



3.0 4.0 6.0 



F'g. 25. Fast fission cross sections vs. neutron energy, illustrating the "threshold" 

nature of the fission reaction, (a) Np 237 . From E. D. Klema, Phys. Rev., 72, 8S 

(1947). (b) Natural uranium; threshold due to U 238 fission (U2). 



Sec. 2C] Interaction with Nuclei 349 

In the very fast- and ultrafast-neutron regions, fission can be induced 
in nuclei of smaller atomic number. Thus, nuclei down to platinum 
have been shown to undergo fission on bombardment by 84-Mev neu- 
trons (K7). 

The fission process has a number of unique features. In addition to 
the relatively tremendous amount of energy released per fission (Q ~ 
200 Mev), the process is also accompanied by the emission of neutrons 
since the fragments, which are comparatively neutron-rich, are emitted 
with sufficient excitation energy to evaporate one or more neutrons. 
It is this feature which has made possible the achievement of a nuclear 
"chain reaction." 

Thermal neutron fission is actually asymmetrical, i.e., the two frag- 
ments have a tendency to be unequal in charge and weight, with the 
peaks in the fragment distribution occurring in the regions of A ~ 100 
and A ~ 140. This tendency toward asymmetry is probably associated 
with the greater stability of nuclei containing the neutron magic num- 
bers 50 and 82 (G7). For fission induced by ultrafast neutrons, on the 
other hand, the fragment distribution appears to be more nearly sym- 
metrical. The change from asymmetry to symmetry with increasing 
bombarding energy is not inconsistent with an explanation in terms of 
the stability of magic number nuclei: a very highly excited nucleus will, 
before undergoing fission, evaporate a considerable number of neutrons; 
it will, therefore, not contain the ~132 neutrons necessary to produce 
nuclei close to two different magic numbers but will, rather, tend to 
split in such a way as to leave both fragments with as close as possible 
to 50 neutrons, i.e., symmetrically. 

5. The (n,2ri) Reaction. The preceding discussion has covered all the 
exoergic reactions, and a few of the endoergic possibilities (inelastic 
scattering, charged particle emission), which can be induced by slow 
and intermediate neutrons. As the neutron energy is increased, through 
the fast, very fast, and ultrafast regions, a number of other threshold 
reactions become possible. One of the most important of these is the 
(n,2n) reaction, whose Q value is the binding energy of a neutron in the 
target nucleus. The product nucleus is an isotope of the target; in many 
cases it is radioactive, frequently decaying by positron emission. 

Neutron binding energies vary throughout the periodic table from 
1.67 Mev in Be 9 to ~20 Mev in C 12 . Table 14 is a compilation of 
(n,2ri) thresholds. Most of the thresholds have been obtained from ob- 
servations on (y,n) reactions which, starting from the same target and 
leading to the same product nucleus, have the same Q values as the 
corresponding (n,2n) reactions. 



350 



The Neutron 



[Pt. VII 



TABLE 14 

Thresholds fob (y,ri) and (n,2ri) Reactions; Neutron Binding Energies 



Target 
Element 


A 


(7,™) 

Threshold t 

(Mev) 


(n,2re) 

Threshold t 

(Mev) 


Product Nucleus 
Half-Life t 


References 


iH 


2 


2.226 ±0.003 


3.34 


stable 




M34 




3 


6.25 ±0.01 


8.33 


stable 




K16 


3L1 


6 


5.35 ±0.20 


6.2 






S26 




7 


7.15 ±0.07 § 


8.2 


stable 




S26 


4 Be 


9 


1.666 ±0.002 


1.85 


5 X 10 


-14 „ 

s a 


M34 


6 B 


10 


8.55 ±0.25 


9.4 






S26 




11 


11.50 ±0.25 


12.6 


stable 




S26 


eC 


12 


' 18.7 ±0.1 § 


20.3 


21m 


3+ 


B6, M13 


?N 


14 


10.65 ±0.2 
10.54 ±0.1 § 


11.3 


10 m 


P + 


M13 
01 


sO 


16 


16.3 ±0.4 


17.3 


2m 


f> + 


B6 


9 F 


19 


10.40 ±0.3 § 


10.9 


1.9h 


H + 


01 


uNa 


23 


12.05 ±0.2 


12.6 


2.6y 


+ 


S26 


liMg 


24 


16.4 ±0.2 


17.1 


12 s 


3 + 


M13, S26 




25 


7.25 ±0.2 


7.5 


stable 




S26 




26 


11.15 ±0.2 


11.6 


stable 




S26 


13AI 


27 


12.75 ±0.2 


13.2 


7s 


fi + 


S26, M13 


i<Si 


28 


16.8 ±0.4 


17.4 


5s 


P + 


M13 




29 


8.45 ±0.2 


8.7 


stable 




S26 


15P 


31 


12.20 ±0.2 


12.6 


25 m 


/?+ 


M13, S26 


ibS 


32 


14.8 ±0.4 


15.3 


3.2s 


+ 


M13 




34 


10.85 ±0.2 


11.2 


stable 




S26 


17CI 


? 


9.95 ±0.2 


10.2 






S26 


19K 


39 


13.2 ±0.2 


13\5 


7.7 m 


3 + 


M13 


2oCa 


40 


15.9 ±0.4 


16.3 


Is 




M13 


22T1 


46 


13.3 ±0.2 


13.6 


3.1h 


f) + 


02 




48? 


11.6 ±0.3 


11.8 


stable 




S26 




49 


8.7 ±0.3 


8.9 


stable 




S26 


23V 


51 


11.15 ±0.2 


11.4 


stable 




S26 


uCv 


50 


13.4 ±0.2 


13.7 


42 m 


B + 


02 




52 


11.80 ±0.25 


12.0 


25 d 


K 


S26 




53 


7.75 ±0.2 


7.9 


stable 


1 


S26 



Sec. 2C] 



Interaction with Nuclei 



351 



TABLE 14 (Continued) 
Thresholds for (y,n) and (n,2n) Reactions; Neutron Binding Energies 



Target 
Element 


A 


Threshold t 
(Mev) 


(n,2n) 
Threshold t 

(Mev) 


Product Nucleus 
Half-Life % 


References 


25MH 


55 


10.1 ±0.2 


10.3 


310 d K 


H35, S26 


26Fe 


34 


13.8 ±0.2 


14.1 


8.9 m /3 + 


M13 




56 


11.15 ±0.25 


11.3 


2.9y 


S26 




57 


7.75 ±0.2 


7.9 


stable 


S26 


27C0 


59 


10.25 ±0.2 


10.5 


72 d, 9.2 h K, + , IT 


S26 


2sNi 


58 


11.7 ±0.2 


11.9 


36 h 0+ 


01 




61? 


7.5 ±0.3 


7.6 


stable 


S26 


29CU 


63 


10.9 ±0.2 


11.1 


10 m /3 + 


M13, S26 






11.2 


11.4 ±0.3 




F40 




65 


10.0 ±0.2 


10.2 


12.8 h + , /3~, K 


M13, S26 


3oZn 


64 


11.7 ±0.2 


11.9 


30 m + 


H35, S26 




66 


11.15 ±0.2 


11.3 


250 d K, j3 + 


S26 




67 


7.00 ±0.2 


7.1 


stable 


S26 




68 


10.15 ±0.2 


10.3 


stable 


S26 




70 


9.20 ±0.2 


9.3 


52 m p~ 


H35 


3iGa 


69 


10.10 ±0.2 


10.2 


68 m (S + , K 


S26 




71 


9.05 ±0.2 


9.2 




S26 


33AS 


75 


10.2 ±0.2 


10.4 


17 d + , /3~, K 


01, S26 


34Se 


82 


9.8 ±0.5 


9.9 


18 m 0~ 


B6 






>9.8 




57 m IT 


B6 




? 


7.30 ±0.2 


7.4 




S26 




? 


9.35 ±0.2 


9.5 




S26 


3sBr 


79 


10.65 ±0.2 


10.9 


6.3 m 3 + 


M13, S26 




81 


10.1 ±0.2 


10.3 


18.5 m pT, P + 


M13, S26 


3sSr 


86 


9.50 ±0.2 


9.6 


65 d K 


S26 




87 


8.40 ±0.2 


8.5 


stable 


S26 




88 


11.15 ±0.2 


11.2 


stable 


S26 


4<)Zr 


90 


12.48 ±0.15 


12.6 


4.5m K01IT 


H35 






12.0 ±0.2 


12.1 


78 h ,8+ 


01 




91 


7.2 ±0.4 


7.3 


stable 


H35 


4iNb 


93 


8.70 ±0.2 


8.8 


10 d /3 - 


S26 


42M0 


92 


13.28 ±0.15 


13.4 


16 m, 17 s (3 + 


H35 




97 


7.1 ±0.3 


7.2 


stable 


H35 




? 


6.75 ±0.25 


6.8 




S26 




? 


7.95 ±0.25 


8.1 




S26 


44RU 


? 


7.05 ±0.2 


7.1 




S26 




? 


9.50 ±0.2 


9.6 




S26 



352 



The Neutron 



[Pt. VII 



TABLE 14 (Continued) 
Thresholds for (y,n) and (w,2n) Reactions; Neutron Binding Energies 



Target 
Element 


A 


(y,n) 

Threshold t 

(Mev)' 


(n,2n) 
Threshold t 

(Mev) 


Product Nucleus 
Half-Life J 


References 


4sRh 


103 


9.35 ±0.2 


9.4 


210 d pr, 0+ 


S26 


4 6 Pd 


? 


7.05 ±0.2 


7.1 




S26 




? 


9.35 ±0.2 


9.4 




S26 


47Ag 


107 


>9.5 




24.5 m 0+ 


B6 




109 


9.05 ±0.2 


9.1 


2.3 m /3 - 


S26, B6 


4«Cd 


113 


6.5 ±0.15 


6.6 


stable 


H35, S26 


4»In 


115 


9.05 ±0.2 


9.1 


50 d IT; 72 a /3 _ , K, 0+ 


S26 


5oSn 


118 


9.10 ±0.2 


9.2 


stable 


S26 




119 


6.55 ±0.15 


6.6 


stable 


H35. S26 




124 


8.50 ±0.15 


8.6 


40 m /3~ 


H35 


5lSb 


121 


9.25 ±0.2 


9.3 


17 m /3 + 


M13 




? 


8.95 ±0.25 


9.0 




S26 




123 


~9.3 


~9.4 


2.8d /3~ 


J4 


52 Te 


? 


6.50 ±0.2 


6.6 




S26 




? 


8.55 ±0.2 


8.6 




S26 


5 si 


127 


9.3 ±0.15 


9.4 


13 d |3 - 


Ol, M13, S26 


55Cs 


133 


9.05 ±0.2 


9.1 


7. Id K 


S26 


56Ba 


J 


6.80 ±0.2 


6.8 




S26 




? 


8.55 ±0.25 


8.6 




S26 


57La 


139 


8.80 ±0.2 


8.9 


stable; 18 h K, IT 


S26 


ssCe 


140 


9.05 ±0.2 


9.1 


140 d K 


S26 




142 


7.15 ±0.2 


7.2 


33 d /3~ 


S26 


69 Pr 


141 


9.40 ±0.10 


9.5 


3.5 m /3 + 


H35 


eoNd 


150 


7.40 ±0.2 


7.4 


2h /3~ 


H35 


7 3 Ta 


181 


7.6 ±0.2 


7.6 


8.2 h (T,K 


M13, S26, J4 


74W 


? 


6.25 ±0.3 


6.3 




S26 




? 


7.15 ±0.3 


7.2 




S26 


75Re 


187 


7.3 ±0.3 


7.3 


93 h 0- K 


S26 


77lr 


193 


7.80 ±0.2 


7.8 


70 d 0- 


S26 


7 S Pt 


194 


9.50 ±0.2 


9.5 


4d K 


S26 




195 


6.1 ±0.1 


6.1 


stable 


P5, S26 




196 


8.20 ±0.2 


8.2 


stable 


S26 


79AU 


197 


8.05 ±0.10 


8.1 


5.6d pr 


P5, H35, S26 



Sec. 2C] 



Interaction with Nuclei 



353 



TABLE 14 (Continued) 
Thresholds for (y,n) and (n,2n) Reactions; Neutron Binding Energies 



Target 
Element 


A 


(t,«) 

Threshold t 

(Mev) 


(n,2n) 

Threshold t 

(Mev) 


Product Nucleus 
Half-Life J 


References 


soHg 


201 


6.25 ±0.2 
6.6 ±0.2 


6.3 
6.6 


stable 


H35 
P5 


siTl 


203 
205 


8.80 ±0.2 
7.5 ±0.15 


8.8 
7.5 


12 d K 

3y 0- 


S26 

H35, S26, P5 


82Pb 


206 
207 
208 


8.25 ±0.10 
6.88 ±0.10 
7.40 ±0.10 


8.3 
6.9 

7.4 


stable 
stable 


PI, P6 

PI, S26, P5 
PI, S26 


83B1 


209 


7.4 ±0.1 


7.4 




M13, P5, S26 


9oTh 


232 


6.35 ±0.04 


6.4 


26 h p~ 


H76, P5 


92U 


238 


5.97 ±0.10 


6.0 


6.8 d 0- 


H76, P5 



t The (n,2n) thresholds are computed from the measured (y,n) thresholds (or vice versa) according 
to the relationship Et(n,2n) = [(A + l)/AlEi(.y,n). 

t The half -lives and radioactivities are from the compilation of Way, Fano, Scott, and Thew (W10). 

§ Thresholds computed from nuclear mass data and used to calibrate the gamma-ray energy scale. 
The most recent data on the thresholds for light elements are summarized by F. Ajzenberg and T. 
Lauritsen, Revs. Modern Phys., 24, 321 (1952). 

In light nuclei, the (n,2n) reaction competes with charged particle 
emission as well as with neutron scattering. In heavy nuclei, charged 
particle reactions are so strongly inhibited by the Coulomb barrier that 
the only important competing reaction — at least, for energies not too 
far above the (n,2n) threshold—is scattering, mainly inelastic at these 
energies. Assuming that this situation prevails, the value of the cross 
section, <x{n,2n), can be computed, given the energy distribution of the 
inelastically scattered neutrons, do(z,E), 



T(n,2n) 






e = E-E t 



d*(z,E) 



(81) 



That is, if the first neutron is emitted with sufficiently low energy 
(s < E — E t ), the residual nucleus will still have enough excitation to 
emit a second neutron and, since neutron emission is the most probable 
mode of decay for heavy nuclei at high excitation energies, it will almost 
always do so. 

At the excitation energies involved, and especially for the low-energy 
part of the spectrum, the distribution of inelastically scattered neutrons 



354 



The Neutron 



[Pt. VII 



is given, to a good approximation, by the Maxwellian distribution 
(Eq. 69). Thus, 



f-E-E, / s \ 

a(n,2n) = a c J \--j e -*T de 



(82) 



= <r c [l - (1 + y)e~»] 
where y = (E — E t )/T. For energies close to the threshold, i.e., y « 1, 



r(n,2ri) 



<fcy" 



2T 2 



{E - E t f 



(82a) 



The (E — E t ) 2 dependence of the (n,2n) cross section near threshold 
has been verified by Fowler and Slye (F40) for the Cu 63 (n,2n) reaction. 




1 


-1 


.35 




1 










" 


.30 






~ 


.25 








.20 


(M 




- 


.15 


s 
fc5" 




- 


.10 








.05 





11.0 



12.0 



13.0 



14.0 



ll.U iz.u la.u 14.U 

Neutron energy (Mey) (Uncertainty of energy scale ± 0.2_Mev) 



Fig. 26. Cross section for the reaction Cu^n^n^Ou 62 in the vicinity of the threshold, 
due to Fowler and Slye (F40). The left-hand scale gives the ratio of the Cu 63 (n,2n) 
cross section to the Cu 66 (n,y) thermal neutron cross section. The absolute cross 
section scale, on the right, is based on (0.56 ± 20%) barns for the thermal neutron 
cross section of Cu 66 in normal copper; it should be increased by ~16 percent (pri- 
vate communication from Fowler). 

Their cross-section measurements for neutron energies between E = E t 
= (11.4 =fc 0.3) Mev and <~14 Mev are shown in Fig. 26. This reaction 
is frequently used as a threshold detector of neutrons of energy E > 
11.5 Mev. 

The C 12 (w,2n) reaction has also been used as a threshold detector for 
very fast neutrons. The cross section from threshold E t = 21 Mev to 
~25 Mev increases rapidly with neutron energy in roughly the expected 
manner (S27). 

The expression for <x(n,2n), Eq. (82), has been derived on the assump- 
tion that the statistical theory, which leads to a Maxwell distribution of 



Sec. 2C] Interaction with Nuclei 355 

the emerging neutrons, is valid in the region of excitation under con- 
sideration. The cross sections near threshold bear out this assumption. 
The statistical theory has been verified over a much wider range of 
excitation energies by the measurements of Bradt and Tendam (B58) 
on the relative cross sections for the (a,n) and (a,2n) reactions on silver 
and rhodium with 15-20 Mev alpha-particles, and by the observations 
of Temmer (T7) and Kelly and Segre (K8) on the excitation functions 
for the (a,ri), (a,2n), and (a,3n) reactions on indium and bismuth with 
alpha-particles up to 40 Mev energy. 

However, application of the statistical theory to (n,2n) reactions 
induced by very fast (and, especially, ultrafast) neutrons is subject to 
the limitation, pointed out by Weisskopf (W17), that neutrons of such 
high energy have a large probability of penetrating through the nuclear 
surface and hence may emerge from the nucleus before they have fully 
shared their energy with the rest of the nuclear constituents. The energy 
distribution of the emerging (first) neutron will, under these circum- 
stances, contain many more fast neutrons than predicted by the evapo- 
ration theory. This situation does not obtain for alpha-particle bom- 
bardment, since the Coulomb barrier impedes the emergence of the 
alpha-particle and helps to achieve the sharing of energy in a true 
compound nucleus. 

In any event, if, after the emission of the first neutron, the residual 
(target) nucleus is still sufficiently excited to emit a neutron, it will do 
so. The energy distribution of the second neutrons is determined by the 
level spacing of the product (final) nucleus at the excitation energy re- 
maining after emission of the second neutron. For incident neutron 
energies not very far above the (n,2ri) threshold, the energy available to 
the second neutron cannot be very large, and the spectrum of second 
neutrons cannot be assumed to follow the Maxwell distribution. In 
essence, the same considerations, which we have applied in the discus- 
sion of the energy distribution of inelastically scattered neutrons, will 
apply in this case. 

6. Multiple-Particle Emission ; Spallation Reactions and Stars. The 
interaction of very fast and ultrafast neutrons with nuclei can be con- 
sidered to take place in two relatively distinct stages: 

(1) The initial interaction of the incident neutron with one or several 
of the nuclear constituents. In this stage the nucleons can be treated as 
relatively independent particles (S21), and the recoiling nucleons (as 
well as the incident neutron) have a non-negligible probability of leav- 
ing the nucleus, carrying off a relatively large fraction of the incident 
energy. 



356 The Neutron [Pt. VII 

(2) Those nucleons which do not directly escape from the nucleus 
will rapidly share their energy among the nuclear constituents. In gen- 
eral, the resulting excited nucleus will have sufficient energy to evapo- 
rate more than one particle. 

The theory of the interaction of neutrons of E ~ 100 Mev with heavy 
nuclei has been developed by Goldberger (Gil). He has computed the 
energy and angular distributions of the emerging nucleons in the first 
stage of the interaction, as well as the distribution of excitation energies 
in the residual nuclei; his computations take into account the velocity 
distribution of the nucleons in the target nucleus (which he treats as a 
Fermi gas confined to the nuclear volume) . 

The process of ejection of nuclear constituents through direct inter- 
action with the incident particle is sometimes referred to as spallation. 1 
The second or evaporation part of the reaction is usually referred to as 
star production, from the characteristic records that such reactions leave 
in nuclear emulsions which are sensitive only to relatively low-energy 
charged particles. 

One method of studying high-energy nuclear interactions is through 
the yields of the various products, usually radioactive (K19). How- 
ever, aside from the obvious limitation of this method (that only radio- 
active products can be detected), it has the disadvantage that the two 
stages of the reaction cannot be separated. 

A more satisfactory method for studying such reactions is by observ- 
ing the emerging particles, either in a cloud chamber or in a sensitive 
nuclear emulsion. In such investigations it is possible to separate the 
two stages of the reaction by the difference in the energy and angular 
distributions of the resulting charged particles. The products of the 
first stage have high energy and forward collimation (Gil). Those of 
the second stage have relatively low energy and are distributed uniformly 
with respect to the direction of the incident particle (L12). The appli- 
cability of the Goldberger model to high-energy nuclear reactions has 
been strikingly verified by the work of Bernardini, Booth, and Linden- 
baum (B19). 

An interesting result of such investigations is the observation of a 
considerable number of high-energy deuterons, emitted in the forward 
direction (H2, B74, Y3, M37). These appear to result from a "pick-up" 
process in which the incident neutron is joined by a nuclear proton, 
which happens to be moving in the same direction and with the proper 
velocity, to form a deuteron (C13, H51, C34). 

1 Spall : A chip or fragment, esp. of stone. Webster's Collegiate Dictionary, fifth 
edition, G. and C. Merriam Co., 1948. 



Sec. 3A] Sources and Detectors 357 

The interaction of ultrafast neutrons with nuclei has been studied in 
cosmic-ray investigations. Neutrons with energies exceeding 10 9 ev are 
present in the cosmic radiation. Many of these investigations have 
employed nuclear emulsions in which cosmic-ray stars are a common 
phenomenon. Le Couteur (LI 2) has shown that the energy distribution 
of the evaporation products in stars can be understood in terms of the 
statistical model; the energy of the emitted protons and alpha-particles 
is given by a modified Maxwell distribution, the modification arising 
from the inhibition of emission of low-energy charged particles by the 
Gamow barrier. 

Owing to the decrease with energy of the primary (n,p) and (n,n) 
cross sections, nuclei are relatively transparent to ultrafast neutrons of 
energy < 300 Mev. However, at neutron energies greater than the 
threshold for meson production (~285 Mev for neutrons on protons or 
neutrons at rest) the reaction cross section is expected to rise again. 
The available evidence indicates that nuclei are essentially opaque (re- 
action cross section « wR 2 ) for neutrons of energy > 10 9 ev. The 
products of the interaction of such neutrons with nuclei are, in the first 
stage, a number (increasing with neutron energy) of mesons (probably 
mostly pions) traveling in the forward direction, and fast nucleons 
(E ~ 100 — 1000 Mev) also in the forward direction; the residual nu- 
cleus, in the second stage, usually has sufficient energy to evaporate a 
large fraction of its particles (F10). 

SECTION 3. SOURCES AND DETECTORS: 
NEUTRON SPECTROSCOPY 

A. Introduction 

To investigate experimentally the properties of the neutron and its 
various interactions, it is necessary to have a neutron source and a neu- 
tron detector. If, in addition, it is desired to confine the investigation to 
neutrons of a specific energy, then either the source must emit mono- 
energetic neutrons or the detector must be sensitive only to neutrons of 
a single energy. The problems of neutron spectroscopy, and the dis- 
cussions of this section, involve the choice of the proper sources or de- 
tectors or both in the various neutron energy ranges. 

The resolution of a given experimental arrangement is a measure of 
the energy spread of the neutrons selected by the source and detector. 
As in other fields, good energy resolution must frequently be paid for 
by decreased intensity, if it can be achieved at all. On the other hand, 
poor resolution is by no means always an unmitigated ill. For example, 
for a comparison of intermediate- and fast-neutron cross sections with 



358 The Neutron [Pt. VII 

the predictions of the continuum theory of Feshbach and Weisskopf, it 
is necessary to average over many levels of the compound nucleus; such 
an average is most easily and accurately achieved by using a source 
whose energy spread is sufficient to cover many levels of the compound 
nucleus. 

The term "resolution" is usually employed in a descriptive, qualita- 
tive connotation, mostly in conjunction with an adjective, such as 
"good" or "poor." The quantitative measure of energy resolution is the 
resolution function, R(E) = S(E)D(E), the product of the source strength 
and the detector efficiency. For many measurements, especially cross- 
section determinations in which there is considerable variation with 
energy (e.g., resonances), a knowledge of R(E) is indispensable for the 
interpretation of the experimental results. A considerable fraction of 
the literature on neutron cross sections is essentially uninterpretable, 
not so much because the experiments were performed with poor resolu- 
tion as because they were performed with sources of unknown energy 
distribution or detectors with unknown efficiency curves. 

However, although this criticism is sometimes applicable to recent 
work, it is certainly not fair to apply it to most of the early investiga- 
tions. To the pioneers in neutron physics, in the pre-World War II 
era, the number of sources and detectors available was severely limited. 
Although many of their cross-section measurements may not have been 
good in the present-day sense, or even quantitatively interpretable, 
their exploratory investigations were invaluable, for only through the 
accumulated knowledge of such explorations were present-day tech- 
niques made possible. In many instances what the early masters lacked 
in technique was more than compensated by the ingenuity of their ex- 
periments and the penetration and insight of their interpretation. 

The first available neutron sources were, naturally, those involved in 
the discovery of the neutron: (a,n) reactions on many elements, espe- 
cially beryllium, using naturally radioactive alpha-emitters. Some of this 
early work has been described in the introductions to Sections 1 and 2. 

The discovery of the photodisintegration of the deuteron by Chad- 
wick and Goldhaber (C8), and of beryllium by Szilard and Chalmers 
(S63) uncovered another source of neutrons, the (y,ri) reaction on these 
elements by gamma-rays from various natural radioactivities. These 
sources differ from the (a,ri) sources in that they provide lower-energy 
neutrons, usually in the intermediate-energy range, since (y,ri) reac- 
tions are endoergic. On the other hand, such sources are usually con- 
siderably weaker than (a,n) sources. However, with the production of 
strong artificially radioactive sources, at first through the use of parti- 
cle accelerators and more recently by neutron irradiation in nuclear re- 



Sec. 3A] Sources and Detectors 359 

actors, a large variety of gamma-ray sources have become available for 
photoneutron production. Furthermore, the development of high- 
energy electron accelerators for the production of intense x-ray beams 
has added another means of intense photoneutron production, although 
these sources are not monoergetic like those produced by a nuclear 
gamma-ray. 

The first extensive investigation of photoneutron production in 
elements other than beryllium and deuterium was made by Bothe and 
Gentner (B52). They utilized the <~17 Mev gamma-rays produced in 
the Li 7 (p,y) reaction, using artificially accelerated protons of 0.4 Mev 
energy. Since then, a large number of investigations of (y,n) reactions 
have been carried out with these and other gamma-ray sources, and with 
x-ray beams from electron accelerators. The results of many of these 
are summarized in Table 14 (Section 2) and in the references therein 
indicated. Of all the nuclei investigated, only beryllium and deuterium 
have photoneutron thresholds of energy less than 6 Mev. 

The development of high-energy heavy-particle accelerators led to the 
discovery of a large number of new and useful neutron-producing reac- 
tions. In addition to providing monoenergetic alpha-particles of energy 
higher than those available from the natural radioactivities, with which 
it was possible to investigate further the (a,n) reactions on beryllium, 
boron, and other elements, it was found possible to utilize other acceler- 
ated nuclei, such as the proton and the deuteron, for neutron production. 

It was soon discovered that (d,ri) reactions have large yields at the 
deuteron energies available from accelerators. Crane, Lauritsen, and 
Soltan (C30) discovered the (d,n) reactions on lithium and beryllium. 
Soon afterward, Lawrence and Livingston (L8) extended the investiga- 
tion of (d,n) reactions to a number of other elements. At about the same 
time (all this in 1933 and 1934) Oliphant, Harteck and Rutherford (03) 
discovered the d + d reaction; this reaction has been of great importance, 
since it provides a strong source of monoenergetic neutrons with com- 
paratively low-energy deuterons, enabling the use of relatively low- 
voltage particle accelerators for neutron sources. Other (d,ri) reactions 
have comparable or higher yields at deuteron energies of a few Mev, 
but the resulting neutrons are heteroenergetic (with the exception of 
the d + t reaction). 

Another reaction type of great importance to neutron spectroscopy is 
the (p,n) reaction. Crane and Lauritsen (C31) discovered the ~Li(p,ri) 
reaction, which has since been extensively used for a monoenergetic neu- 
tron source. However, this reaction does not really yield monoenergetic 
neutrons, since it also gives rise to a second neutron group, due to an 
excited state of Be 7 , 435 kev above the ground state. Fortunately, the 



360 The Neutron [Pt. VII 

second group has a comparatively low yield (<10 percent) for protons 
up to -~4 Mev. A number of other (p,ri) reactions have been studied 
and used from time to time for neutron sources, but none so extensively 
as the Li 7 (p,n) reaction. 

All the above-mentioned neutron sources yield energies in the inter- 
mediate, fast, and very fast ranges. Until the advent of ultrahigh- 
energy accelerators, the only source of neutrons in the ultrafast-energy 
range was from the cosmic radiation. The presence of high-energy 
neutrons as a component of cosmic rays was established by the experi- 
ments of Rumbaugh and Locher (R25) and of Fiinfer (F51), and has 
been used by many investigators to study the nuclear interactions in- 
duced by ultrafast neutrons. Although particle accelerators are at 
present capable of producing neutrons of energy up to ~400 Mev, and 
higher-energy accelerators (a few Bev) are now being constructed, cos- 
mic radiation still remains the only source in immediate prospect for 
neutrons of energies of, say, > 10 Bev. 

At the other extreme, the slow-neutron region has been most exten- 
sively investigated. Sources of slow neutrons almost invariably have 
their origin in fast neutrons, slowed down in paraffin or some other 
material containing light elements. Since such sources yield a broad 
distribution of neutron energies, it is necessary to employ some form of 
neutron monochromator for studies requiring monoenergetic neutrons. 
A number of monochromators have been, and are being, developed, and 
their effective range has been slowly pushed up toward the intermediate- 
energy region, so that there now remains only a small gap between the 
monoenergetic neutrons available from charged particle reactions and 
from slow-neutron monochromators. 1 The availability of very great 
neutron intensities from nuclear reactors has provided a great impetus 
to the development of more effective neutron monochromators of greater 
range and flexibility. 

The development of neutron detectors has rapidly followed the exten- 
sion of knowledge of neutron reactions; practically every new discovery 
has led to a new means of neutron detection. Thus, the observation of 
proton recoils by Curie and Joliot (C37), made even before the identifi- 
cation of the neutron as a new particle, led to the technique of observing 
neutrons in ionization chambers, electroscopes, and cloud chambers by 
lining these instruments with paraffin. The discovery of charged parti- 
cle reactions in lithium, boron, and nitrogen enabled the detection of 

1 Actually, the gap is being closed from the intermediate-energy end as well. Thus, 
Hibdon, Langsdorf, and Holland [Phys. Rev., 85, 595 (1952)] have succeeded in 
studying the 2-25 kev range with an energy resolution of 2 kev, using the Li(p,w) 
reaction. 



Sec. 3B] Sources and Detectors 361 

neutrons through the incorporation of these substances in ionization 
chambers, proportional counters, and cloud chambers. These and other 
reactions, as well as proton recoils, can also be observed in nuclear 
emulsions. 

The discovery of neutron-induced radioactivity provided still another 
means of neutron detection which could be used to investigate specific 
energies or energy ranges — thermal neutrons through 1/v capture cross 
sections, specific slow-neutron energies through various resonances, 
fast neutrons by means of threshold reactions. As neutron reactions 
have been further understood, and as various techniques of charged 
particle counting have been improved and extended, the variety of neu- 
tron detectors has increased until now it is possible to find a suitable 
neutron detector at almost any energy. 

This is not to say that detector problems are negligible in neutron 
studies; as is so often the case, ease of detection is in direct proportion to 
the available intensity, so that the source and detector problems of 
neutron spectroscopy go hand in hand. Especially in the fast-, very 
fast-, and ultrafast-neutron energy ranges, detectors are of relatively low 
efficiency, and available sources are never quite strong enough, so that 
neutron spectroscopy in these energy regions still presents difficult 
problems. 

Nevertheless, it seems fair to summarize by saying that the available 
techniques of neutron spectroscopy allow an almost complete coverage 
of the range of energies from to ~300 Mev with relatively few signifi- 
cant gaps, and that, with foreseeable extensions of available techniques, 
the existing gaps should soon be closed. 

B. Neutron Sources 

1. Radioactive (a,ri) Sources. The discovery of the neutron involved 
the reaction 

4 Be 9 + 2 He 4 -» 6 C 12 + on 1 

induced by bombarding beryllium with a-particles emitted by the natu- 
rally radioactive elements. Although many other neutron-producing 
reactions have since been discovered, the above reaction is still the basis 
for some of the most extensively used neutron sources. 

According to available mass values (T16, B2), this reaction is exoergic, 
with a Q value of 5.65 Mev. Thus, starting with the polonium alpha- 
particles (energy 5.30 Mev), the emergent neutrons should have a spread 
of energies between 10.8 Mev (outgoing neutron in the same direction 
as the incoming alpha) and 6.7 Mev (outgoing neutron in the opposite 
direction from the incoming alpha). 



362 The Neutron [Pt. VII 

However, the neutrons observed in the bombardment of beryllium 
with polonium alpha-particles have a considerably greater energy 
spread, ranging from the above maximum to energies well below 1 Mev. 
The observed energy spread arises from two causes: (1) In the above 
reaction, the C 12 nucleus may sometimes be left in an excited state, re- 
sulting in less available energy for the outgoing neutron. This possi- 
bility will, for monoenergetic incident alpha-particles, result in the 
appearance of groups in the spectrum of the outgoing neutrons, each 
group corresponding to an excited state of the C 12 nucleus. Evidence 
for the existence and energy values of the C 12 levels is summarized by 
Hornyak, Lauritsen, Morrison, and Fowler (H67). (2) In most (a,n) 
sources, the thickness of the beryllium target is large compared to the 
range of the impinging alpha-particles. (The range of a polonium 
alpha-particle is 3.66 cm in standard air.) Since the cross section for 
the neutron-producing reaction is small compared to the cross section 
for energy loss by collisions with atomic electrons, very few nuclear 
processes occur while the alpha-particle has its full, initial energy. 
Thus, even if all the reactions led to the ground state of C 12 , the out- 
going neutrons in the forward direction would have an energy spread 
ranging from the maximum (10.8 Mev) down to 5.2 Mev (correspond- 
ing to zero incident alpha-particle energy). 

The energy spectrum of Po-a-Be neutrons is further complicated as a 
result of the variation of the reaction cross section with the incident 
alpha-particle energy. Thus, the necessity for the alpha-particle to 
penetrate through the potential barrier of the beryllium nucleus de- 
creases the neutron yield for low-energy alpha-particles; the height of 
the potential barrier is ~3.7 Mev. Furthermore, the level structure of 
the compound nucleus, C 13 , leads to resonances in the cross section 
(H67). 

As a result of the effects discussed above, the neutron spectrum from 
a (thick target) Po-a-Be source is complex, and cannot be predicted in 
detail. A number of attempts have been made (A27) to measure the 
neutron spectrum from such a source using proton recoils, in nuclear 
emulsions, as a neutron detector. The result of a recent measurement, 
due to Whitmore and Baker (W20), is shown in Fig. 27. 

Among the alpha-emitting radioactive elements, polonium is compara- 
tively difficult to obtain in quantities sufficient to produce strong neu- 
tron sources. The elements radium and radon are, however, commer- 
cially available in sufficient quantity so that they are most frequently 
used for neutron sources. Radium has the advantage of a very long half- 
life (~1600 years, as compared to 3.825 days for radon and 138 days for 
polonium), which makes it particularly suitable for long-lived sources. 



Sec. 3B] 



Sources and Detectors 



363 



On the other hand, polonium and, even more so, radon require a con- 
siderably smaller mass of beryllium in the source mixture to approxi- 
mate a thick target (since the mass per unit radioactive strength of a 
radioactive element is proportional to its half-life). In particular, 
radon requires quite small quantities of beryllium for high neutron 
yields, and presents no difficulty in mixing, since it is a noble gas and 
diffuses uniformly through powdered beryllium; thus, Rn-a-Be sources 









\ 


























V- 


























T N 









4 5 6 7 

Neutron Energy (Mev) 



10 



11 



Fig. 27. Energy distribution of neutrons from a Po-a-Be source (W20). 



can be made quite small. However, owing to the inconvenience of 
working with a gas of such short half-life, radon is now seldom used for 
neutron sources. 

Po-a-Be sources have the added advantage that in the decay of 
polonium there is a comparatively negligible gamma-ray emission, 
which makes the handling of such sources relatively simple. The decay 
products of radium, on the other hand, emit a prodigious quantity of 
gamma-radiation, and suitable precautions for protection against these 
radiations must be observed in the handling of these sources. Never- 
theless, owing to the conveniences of availability and long life, Ra-a-Be 
neutron sources are very widely used. 

Another disadvantage of polonium as compared to radium is that it is 
more difficult to manipulate, despite the absence of gamma-radiation. 
One method of preparation of a Po-a-Be source, described by Spinks and 
Graham (S49), consists in sandwiching a platinum foil, on which the 
polonium is deposited, inside a cylinder of beryllium. Such a source 



364 The Neutron [Pt. VII 

has a smaller yield per curie (~3^) than an intimate mixture, and the 
neutron emission is not isotropic. 

During World War II considerable experience was obtained in the 
preparation of radium-beryllium mixtures. 1 The preparation and 
handling of such sources has been described by Anderson and Feld 
(A26) and, in greater detail, by Anderson (A27). Most of these con- 
sist of an intimate, physical mixture of radium bromide and beryllium 
metal powder, pressed into pellets of density ~1.75 g/cm 3 . In addition 
to their small size (thereby more closely approximating a point source), 
pressed sources are more likely to remain constant in time because of 
the greater physical stability of the mixture. 

The neutron spectrum from Ra-a-Be sources is even more complex 
than that of Po-a-Be, owing to the variety of alpha-particles emitted 
by radium and its decay products, as shown in Table 15. The presence 
of polonium alpha-particles in a Ra-a-Be source is governed by the decay 
of radium D, with a 22-year half-life. Hence, in relatively young (a few 
years old) sources, this last alpha-particle is not appreciably present. 

TABLE 15 
Alpha-Particle Energies from Radium and Its Decay Products 



Alpha-Emitter 


Half-Life 


Energy (Mev) 


Ra 


1620 y 


J 4.795 (93.5%) 
14.611 (6.5%) 




Rn 


3.825 d 


5.486 


RaA 


3.05 m 


5.998 


RaC 


1.5 X 10- 4 s 


7.680 


RaF(Po) 


138.3 d 


5.300 



The rest of the alpha-particles are fully present after a few weeks. All 
the following discussion (spectrum, yield) is concerned with such young 
sources. Since both the spectrum and yield of Po-a-Be sources are 
comparatively well known, it is easy to take into account the changes 
in the source due to the accumulation of polonium. 

The Ra-a-Be neutron spectrum has not been nearly so extensively 
investigated as Po-a-Be, mainly because of the difficulties of neutron 
measurement in the accompanying high gamma-ray background. Its 
properties may be roughly summarized as follows (A27): The fast- 
neutron spectrum extends to a maximum energy of ~13 Mev, with a 
broad peak at ~4 Mev. There appears to be a substantial group of 
intermediate-energy neutrons, but there is considerable uncertainty as 

1 A large number of these sources were prepared in the laboratory of the Radium 
Chemical Company, 570 Lexington Ave., New York City. 



Sec. 3B] Sources and Detectors 365 

to their amount and energy. Various estimates of the yield of inter- 
mediate neutrons range from ~10-30 percent of the total yield. These 
have been ascribed to the reaction 

He 4 + Be 9 -* 3He 4 + n 1 

(in which a number of intermediate steps have been omitted). The 
low-energy group may also arise, in some part, from a (y,n) reaction on 
beryllium. In any event, the low-energy group does not seem to be 
present in Po-a-Be sources, owing either to the absence of higher-energy 
alpha-particles (those from RaC are assumed mainly responsible for 
the 3a reaction) or of gamma-rays, or both. 

Other light elements beside beryllium can be used to produce neutrons 
through (a,n) reactions. Thus, a pressed Ra-a-B source has been pre- 
pared, and its spectrum and yield studied (A27). Both B 10 and B 11 
undergo exoergic (a,n) reactions (Q = 1.18 and 0.28 Mev, respectively), 
the latter being responsible for most of the neutron yield. The spectrum 
is comparatively simpler than that of a Ra-a-Be source, rising rapidly to 
a maximum at ~3 Mev, and then falling rapidly to zero at ~6 Mev. 
There does not appear to be any appreciable intermediate-energy 
component. 

The reaction F 19 (a,n) is believed (from mass values) to be slightly 
exoergic, by <0.5 Mev. Bretscher, Cook, Martin, and Wilkinson 
(B62) have prepared a source composed of the relatively stable com- 
plex, RaBeF 4 , which they suggest for a standard neutron source, since 
the characteristics of the complex are not expected to change appre- 
ciably with time. 

The yields of the various sources discussed above have been studied 
by a number of methods (A27). In general, for a given mixture of 

TABLE 16 

Yields or Radioactive (a,n) Sources fob Intimate Mixtures of 
an Alpha-Emitter and a Neutron-Producing Material 

Source Y (10 6 neutrons/curie • sec) 
Ra-a-Be 17 

Rn-a-Be 15 

Ra-a-B 6.8 

Po-a-Be 3 

RaBeF* 2.53 f 

t Since the mixture is fixed, the value given is that of an actual source com- 
posed of the complex. This yield could be improved by adding beryllium, but 
this would nullify the purpose of such a source, namely, the elimination of pos- 
sible changes due to alteration of the physical composition. 



366 



The Neutron 



[Pt. VII 



some alpha-emitting compound (a-mat) and neutron-producing material 
(X), the yield is given by the relationship 



Y=Y 



M(X) 



M(X) + M(a-mat) 



(83) 



Values of F (yield for an <x> ratio of X to a-mat) are given in Table 16. 
(<x,ri) yields from various materials have been extensively investi- 
gated, and are summarized by Anderson (A27). A number of these in- 
vestigations have employed thin sources and targets (S61, W3, H9), 
while others have measured thick target yields. Most of the investi- 
gators employed polonium alpha-particles, varying their energy by 
changing the pressure of gas between source and detector. The thin 
target (a,n) cross section for beryllium (due to Halpern, H9) is shown 
in Fig. 28a. Figure 28b shows the results of Segre and Wiegand on the 
thick target yields of beryllium, boron, and fluorine. Thick target 
yields, for (artificially accelerated) 9-Mev alpha-particles, have been 
measured by Ridenour and Henderson (Rll), and for 30-Mev alpha- 



TABLE 17 
Neutron Yields fob Polonium Alpha-Particles on Thick Targets 





Yield per 10 6 Alphas 


Element 














Roberts 


Segre and 
Wiegand 

(A27) 


Walker 


Halpern 


Szalay 




(A27) 


(W3) 


(H9) 


(S61) 


Li 


2.6 


4.7 








Be 


80 


73 




50 




B 


24 


19 


19 






C 


0.11 










N 


0.01 













0.07 










F 


12 


10 








Na 


1.5 










Mg 


1.4 






0.5 




Al 


0.74 






0.25 


0.22 


Si 


0.16 










CI 


0.11 










A 


0.38 











Sec. 3B] 



Sources and Detectors 



367 



0.2 



0.6 



1.0 



a range in air (cm) 
1.4 1.8 2.2 2.6 



3.0 



3.4 



3.8 



0.44 
0.40 
0.36 

"£•0.32 

a 

a 0.28 

x> 

"^0.24 

o 

'•§ 0.20 

o> 

So.16 

CO 

J 0.121- 
0.08 
0.04 




I I j 1 

- Be 


' 1 ■] ! -1 


l ■ i -v — 1 


i | ( I- " 


- 




• •M 
• • 

• 
• 
• 

• 
• 


• • 

• 

• 
• 








• 




- 

• 


• • 

• 


• 
• 


• 
• 




- • 










- ••)• i i 


1 1 


1 


1 1 1 


1 



0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 
Energy (Mev) 

(a) 



4.5 



5.0 



80 
70 

Jso 

a, 
J 50 













































fl 






































































































1 Beryllium 

2 Boron 

3 Fluorine 














































































































































































































































































































































































































'2 






















































































3 















































2 3 

Energy (Mev) 



(b) 

Fig. 28. (a,n) yields vs. neutron energy for a number of elements, (a) Thin target 
(a,n) cross section of beryllium, due to Halpern (H9). (b) Thick target (a,n) excita- 
tion functions of beryllium, boron, and fluorine, due to Segre and Wiegand (A27). 



368 The Neutron [Pt. VII 

particles by Allen, Nechaj, Sun, and Jennings (A8), who also measured 
the angular distributions of the neutrons. The results of a number of 
investigations of thick target yields from various elements for polonium 
alpha-particles are summarized in Table 17, in which the data of Rob- 
erts and of Segre and Wiegand are taken from Anderson's summary 
(A27). The results of the last three investigators are obtained by inte- 
gration of their thin target yield curves. 

2. Photoneutron Sources. Photons can cause neutron emission from 
nuclei, provided that their energy is greater than the neutron binding 
energy. In so far as the (y,n) reaction is concerned, the periodic table 
is conveniently divided into two groups: (1) deuterium and beryllium, 
and (2) all the rest. The former have (y,n) thresholds of (2.226 ± 0.003) 
and (1.666 ± 0.002) Mev, respectively. The latter have thresholds in 
excess of 6 Mev. Since no radioactive nuclei which emit gamma-rays of 
such high energy are known, radioactive (y,n) sources employ only 
beryllium or deuterium. The (prompt) gamma-rays from various nu- 
clear reactions or the x-rays obtainable from high-energy electron 
accelerators can be used to obtain neutrons from other nuclei as well. 

(a) Radioactive (y,n) Sources with Deuterium and Beryllium. The 
bombardment of deuterium or beryllium by monoenergetic gamma-rays 
results, for a given angle 6 between the emitted neutron and incident 
gamma-ray, in monoenergetic neutrons, according to the relation- 
ship (W8) 



B ' - (^T 2 ) 



ET 2 

E y -Q- 



5^E y 



1862(A - 1) 
2(A - l)(E y - Q) 
931 A 3 



+ 5 cos 6 (84a) 



(84b) 



E n is the neutron energy; E y , the gamma-ray energy; Q, the neutron 
binding energy, all in Mev. A is the mass number of the target nucleus. 
In almost all practical photoneutron sources, for reasons of intensity, 
the gamma-ray source is surrounded by beryllium or deuterium, so that 
the neutrons have an inherent energy spread corresponding to an iso- 
tropic distribution in the angle 0, 

AE n = 25 (84c) 

For 100-kev neutrons, AE n /E n « 4 percent for beryllium, and « 25 
percent for deuterium. The relative spread decreases with increasing 
neutron energy. 

The above is, however, not the main cause of energy spread. For 
most sources, a larger uncertainty in the neutron energy arises from the 



Sec. 3B] 



Sources and Detectors 



369 



fact that considerable quantities of beryllium or deuterium must be used 
to obtain usable neutron intensities. Since both beryllium and deuter- 
ium are quite light, and since the neutrons have a non-negligible proba- 
bility of undergoing a scattering before emerging from the source, an 
appreciable energy spread may thereby be introduced. In addition, 
neutron scattering in the source has the effect of reducing the mean 
energy of the emerging neutrons. Furthermore, the gamma-rays can 
lose energy by Compton scattering and then produce neutrons, an 
effect which introduces a further uncertainty in the neutron energy. 
Since the Compton cross section is ~10 3 times that for photodisinte- 
gration, a source containing an infinite quantity of beryllium or deuter- 
ium will produce only ~1 neutron per 1000 gamma-rays. 

TABLE 18 
Photoneutbon Soukces 











Standard 










Yield | 


Source 


Tl A 


£ T (Mev) 


E n (Mev) 


(10 4 neutrons/ 

sec • curie) 

(1 gram at 

1 em) 


Na 24 + Be 


14. 8h 


2.76 


0.83 


13 


Na 24 + D 2 




2.76 


0.22 


27 


Mn 66 + Be 


2.59h 


1.81,2.13,2.7 


0.15,0.30 


2.9 


Mn 66 + D 2 




2.7 


0.22 


0.31 


Ga 72 + Be 


14. lh 


1.87,2.21,2.51 


(0.78) f 


5 


Ga 72 + D 2 




2.51 


0.13 


6 


Y 88 + Be 


87 d 


1.9,2.8 


0.158 ± 0.005 § 


10 


Y 88 +D 




2.8 


(0.31) 


0.3 


In 116 + Be 


54 m 


1.8,2.1 


0.30 


0.82 


Sb 124 + Be 


60 d 


1.7 


0.024 ±0.003 § 


19 


La 140 + Be 


40 d 


2.50 


0.62 


0.3 


La 140 + D 2 




2.50 


0.151 ± 0.008 § 


0.8 


MsTh + Be 


6.7y 


1.80, 2.62 


0.827 ± 0.030 § 


3.5 


MsTh + D 2 




2.62 (ThC") 


0. 197 ± 0.010 § 


9.5 


Ra + Be 


1620 y 


1.69, 1.15,1.82, 
2.09, 2.20, 0.4# 


a mess 


3.0 


Ra + D 2 




2.42 


0.12 


0.1 



t The standard yield is taken to be that of 1 gram of beryllium or heavy water 
at 1 cm from 1 curie of the substance indicated. 
% Values in parentheses are estimates. 
§ Due to Hanson (H33). 



370 The Neutron [Pt. VII 

These factors, as well as other aspects of the production and use of 
photoneutron sources, are discussed in considerable detail by Watten- 
berg (W8,W7). Table 18 summarizes the properties of available radio- 
active (y,n) sources. 

The characteristics of the cross section for the D(-y,n) reaction have 
been discussed in Section 1. Considerably less is known concerning the 
Be(y,n) cross section. The available evidence, both experimental (R26) 




Fig. 29. Design of the primary photoneutron source standard at the National 

Bureau of Standards (C42). The beryllium sphere is 4 cm in diameter and holds, 

at the center (C), a 1-gram capsule of radium. 

and theoretical (G32), indicates that the cross section passes through 
at least one maximum and one minimum as the gamma-ray energy is 
increased from threshold to 2.76 Mev. 

Prior to the extensive availability of strong artificially produced 
gamma-ray sources, Ra-y-Be sources were extensively used to provide 
intermediate-energy neutrons. Following a suggestion of Gamerts- 
f elder and Goldhaber (G2), such a source has been prepared by Curtiss 
and Carson (C42) at the National Bureau of Standards, to serve as a 
permanent neutron standard. Since a Ra-7-Be source does not require 
mixing of the radium and beryllium (with the attendant possibilities 
for physical change), its neutron output should not vary with time. In 
the standard source, a pressed radium bromide pellet is placed at the 
center of a carefully machined sphere of beryllium metal; the design 
of the standard source is shown in Fig. 29. 

The absolute yield of such a source could be obtained, without the 
necessity of any neutron measurement, by a method developed by 



Sec. 3B] Sources and Detectors 371 

Paneth and Gluckauf (P2, G8). They measure the total accumulation 
of helium 1 after a known time of irradiation of the beryllium. 

An interesting application of the photodisintegration process is as a 
gamma-ray detector which is completely insensitive to gamma-rays of 
energy below the photodisintegration threshold (P8a). Myers and Wat- 
tenberg (M42) have used this device to detect the presence of a small 
component of "cross-over" gamma-rays when two or more gamma-rays, 
both of which are below the photodisintegration threshold of beryllium 
or deuterium, are emitted in cascade. 

(b) Photoneutrons from High-Energy Gamma-Rays and X-Rays. Pho- 
toneutron reactions with high-energy gamma-rays have been investi- 
gated in a large number of elements. Following the work of Bothe and 
Gentner (B52), the ~17-Mev Li(p,y) gamma-rays have been used to 
study the (y,ri) cross sections of many elements, by Waffler and Hirzel 
(Wl) and by McDaniel, Walker, and Stearns (M12). However, the 
most extensive investigations have employed x-ray beams from electron 
accelerators, mainly betatrons. 

The x-rays are produced by causing the accelerated electrons to strike 
a target, usually of some heavy element. The resulting radiation is 
allowed to fall on the material under investigation. (7,71) reactions are 
detected either through direct observation of the neutrons, or by de- 
tection of the radiations from the product (usually /3 + -radioactive) 
nuclei. 

The shape of the x-ray spectrum from an electron accelerator depends 
on the target thickness. For relatively thin targets, the distribution of 
x-ray quanta follows a bremsstrahlung spectrum, at least for energies 
not too far below the maximum (electron) energy, 

dN(E y ) = —1 (85) 

Ey 

The measurement of (y,n) cross sections with such x-ray beams involves 
the complication of dealing with a heteroenergetic source. However, if 
the electron energy can be varied, (y,n) yields can be measured as a func- 
tion of the maximum x-ray energy. The results of three such studies, 
due to Diven and Almy (D15), are shown in Fig. 30a. Such curves can 
be interpreted in terms of the (y,n) cross section vs. E y , provided that 
the x-ray spectrum is known and the x-ray intensity (the value of k in 
Eq. 85) is calibrated. 

1 The reaction is Be 9 (7,n)Be 8 ; Be 8 -> 2He 4 . 



372 



The Neutron 



[Pt. VII 



Some curves of <r(y,ri) vs. E y are shown in Fig. 30b. The striking fea- 
ture of such cross-section curves, first noted by Baldwin and Klaiber 
(B7), is the strong resonance shape. This shape has been observed for 
all the nuclei studied, although the positions of the maxima and the 



16 



14 



12 



za 

a 
g 8 

B 



a) 





















































































































































































































































































































Ag 1 


07,109 






Cu 63 




/A 


J 27 

































































































































































10 12 14 16 18 20 

Maximum x-ray energy (Mev) • 



22 



24 



Fig. 30a. Relative neutron yields vs. maximum x-ray energy for three nuclei, due 
to Diven and Almy (D15). The ordinates are in arbitrary units. 



resonance widths vary from nucleus to nucleus (D15, M13). Particu- 
larly accurate work, determining the resonance constants for many 
nuclei, has been done by the Saskatchewan Group (J4, K4). 

An explanation of these resonances has been advanced by Goldhaber 
and Teller (G16). They have postulated the possibility of dipole vibra- 
tions in which the protons (as a whole) oscillate with respect to the neu- 
trons in the nucleus. The general features of the process of dipole radia- 
tion capture have been derived by Levinger and Bethe (LI 6). 



Sec. 3B] 



Sources and Detectors 



373 



Neutron yields, due to the (y,n) reaction, have been measured for 
many elements and at many x-ray energies, up to 330 Mev (P31, T8). 
Neutron sources of considerable strength can be obtained, through 
(y,n) reactions, from electron accelerators. Thus, by using the elec- 
trons from a 3.2-Mev linear accelerator to produce gamma-rays in a 
lead target, and by irradiating a heavy water (or beryllium) target with 
these gamma-rays, Cockroft, Duckworth, and Merrison (C16) obtained 



































































































































































w 
































































































































































































































































Ta 




















































































































































p\ 


3u 6 


3 









































































































































W 11 12 13 14 15 16 17 18 19 20 21 

7 -ray energy (Mev) 

Fig. 30b. (y,n) cross sections vs. 7-ray energy for copper, silver, and tantalum 

(D15.M13). 

average neutron yields of ~10 9 neutrons/sec and peak yields of ~2 
X 10 12 neutrons/sec (in ~2 ^sec bursts). Calculations based on the 
curves of Fig. 30 indicate that an electron beam of 1 /tamp (average) 
and 20 Mev energy, impinging directly on a thick target of a medium 
or heavy element, should yield ~10 10 neutrons/sec. 

The spectrum of neutron energies resulting from x-ray-induced (y,n) 
reactions is relatively broad, corresponding to "evaporation" from an 
excited compound nucleus; the theory of the evaporation process, and 
of its Z-dependence, has been discussed by Heidmann and Bethe (H52). 
The Maxwellian form of the neutron energy distribution has been veri- 
fied by Byerly and Stephens (B80). However, there is expected to be a 
high-energy "tail" of neutrons ejected by direct gamma-ray action, with 
a non-spherical angular distribution (C29). 



374 The Neutron [Pt. VII 

3. Accelerated Charged Particle Sources. Radioactive alpha-particle 
and gamma-ray neutron sources are of rather limited usefulness, espe- 
cially for the production of intense beams of monoenergetic neutrons. 
The availability of Van de Graaff and cyclotron accelerators, capable 
of delivering strong monoenergetic currents of various nuclear projec- 
tiles, makes possible the production of strong neutron sources by a 
variety of nuclear reactions. These projectiles include protons (p), 
deuterons (d), tritons (t), alpha-particles (a), and heavier nuclei. In 
this section we shall consider only reactions induced by the first three. 
[(a,n) reactions have been discussed in connection with radioactive 
neutron sources.] 

We shall mainly emphasize, in this section, reactions which can lead 
to monoenergetic neutrons. This possibility exists whenever the energy 
of the first excited state of the product nucleus is too great to be excited 
by the projectiles used. Depending on the nucleus involved, the first 
excited state may lie anywhere from a few hundred kev to many Mev 
above the ground state. In some cases, sources may be effectively mono- 
energetic even when the first excited state is energetically available, 
owing to a relatively low yield of excited product nuclei. 

With the available reactions which yield monoenergetic neutrons, it 
appears possible to cover the neutron energy range from a few kev to 
20 Mev. To cover the rest of the very fast- and the ultrafast-neutron 
ranges, it is necessary to resort to heteroenergetic sources, and to de- 
pend on the detector for any sorting out of the neutron energies. Prior 
to the availability of tritium, and the use of the t-d reaction, heteroener- 
getic (d,n) reactions were the only ones available for obtaining very 
fast neutrons. Such sources will also be considered in this discussion. 

Given the requisite reaction, a strong current of monoenergetic 
projectiles, and a thin target, the neutrons emerging at a given angle 
with respect to the projectile direction have a definite energy. The rela- 
tionship between the neutron energy E n , the angle 8, the projectile 
energy E i} the reaction Q value, and the masses of the particles involved 
has been frequently described. 1 The properties of monoenergetic neu- 
tron sources are collected in a review article by Hanson, Taschek, and 
Williams (H34), upon which this summary leans heavily. The energy- 
angle relationships in a given reaction are conveniently represented in 
the form of a nomograph, developed by McKibben (M15), 2 of which 
some examples will be shown in the following. 

1 See, for example, Part VI. 

2 Such nomographs for the most extensively used reactions can be purchased as 
document MDDC 223 from the Document Division of the AEC, Oak Ridge, Ten- 
nessee. They have been reprinted in the article of Hanson, Taschek, and Williams 
(H34). 



Sec. 3B] Sources and Detectors 375 

A McKibben nomograph (Figs. 31, 33, 37, 40) consists of two sets of 
semicircles (solid and broken) and two sets of radial lines. The solid 
semicircles, centered at the origin of the (lower) neutron energy scale, 
represent various values of the neutron energy in the laboratory coordi- 
nate system. The solid radial lines, from the same origin, represent 
angles of emission of the neutrons in the laboratory system. The 
broken semicircles (which are not concentric) represent various energies 
of bombarding particle (p or d) ; the bombarding energy is given by the 
intersection of a broken semicircle with the upper of the two horizontal 
axes. The broken radial lines represent loci of equal values of the angle 
of neutron emission in the center-of-mass coordinate system. 

Thus, given a value of the bombarding particle energy, the energies 
of the emitted neutrons are given by the intersections of the appropriate 
broken semicircle and the solid semicircles; to each neutron energy (in- 
tersection) there corresponds a definite laboratory angle (solid line) and 
a definite center-of-mass angle (broken line). 

(a) Intermediate and Fast Neutrons from (p,n) Reactions. These reac- 
tions are endoergic. Hence, by bombarding thin targets with protons 
of energy only slightly above threshold it is possible to obtain monoener- 
getic neutrons of relatively low energy. The minimum energy obtain- 
able from such sources is limited by the fact that, at proton energies 
only slightly above threshold, there are two neutron energies correspond- 
ing to each angle of emergence. (Neutrons emitted at forward and back- 
ward angles in the center-of-mass system all lie within a cone of apex 
angle <180° in the laboratory system.) However, as soon as the apex 
angle of the cone of neutron emergence becomes 180°, the energy-angle 
relationship is unique. In this respect, the heavier the target nucleus, 
the lower the energy at which the neutrons for a given angle are mono- 
energetic. However, the necessity for penetration, by the proton, of 
the Coulomb barrier limits the possible target nuclei to low Z (<25). 

Some properties of known {p,n) reactions are summarized in Table 19. 
Most of these data are from the work of Richards and co-workers at the 
University of Wisconsin, and have been collected by Richards, Smith, 
and Browne (RIO), who give references to the original investigations. 
The minimum neutron energy at threshold (fifth column) arises from 
the center-of-mass motion of the system. 

E t 
E - mi ° = CA+I) 5 (86) 

where A is the mass number of the target nucleus. The sixth column 
gives the minimum energy of monoenergetic neutrons in the forward 



376 



The Neutron 



[Pt. VII 



TABLE 19 
Properties op (p,n) Reactions for Z < 25 (Mostly from RIO) 



Target 


Product 


Observed 
E, (Mev) 


-Q 

(Mev) 


■^w.min 

(kev) 


-^ re.min 

at0° 
(kev) 


Lowest 
Level 
(Mev) 


xH 2 


21H 1 


3.339 ±0.015 


2.225 


371 


1979 




iH« 


2 He 3 


1.019 ±0.001 


0.764 


63.7 


286.5 


>2.5 


3 Li 7 


4 Be 7 


1.882 ±0.002 


1.646 


29.4 


120.1 


0.435 


4 Be 9 


5B 9 


2.059 ±0.002 


1.852 


20.6 


83.4 


>1.5 


6 B" 


eC 11 


3.015 ±0.003 


2.762 


20.9 


84.5 


2.02 


6 C 12 


6 N 12 


20.0 ±0.1 


18.5 


118 


477 




eC 13 


6 N 13 


3.236 ± 0.003 


3.003 


16.5 


66.4 


2.383 


6 C 14 


6 N" 


0.664 ±0.009 


0.620 


2.9 


11.8 


2.3 


8 18 


8 F 18 


2.590 ±0.004 


2.453 


7.2 


28.8 




9 F 19 


ioNe 19 


4.18 ±0.25 


3.97 


10.5 


42 




nNa 23 


i 2 Mg 23 


4.78 ±0.3 


4.58 


8.3 


33 




itCI" 


isA 37 


1.640 ±0.004 


1.598 


1.1 


4.6 


1.4 


18 A«> 


19K* 


<2.4(?) 


2.3(?) 


1.6 


5.7 


0.81 


i»K« 


2oCa 41 


1.25 ±0.02 


1.22 


0.7 


2.8 


1.95 


2lSc 45 


22 Ti 45 


~2.85 


2.79 


1.35 


5.4 




2 3 V 61 


2 4Cr» 


1.562 ±0.006 


1.532 


0.58 


2.3 


0.775 f 


25 Mn 55 


26 Fe 66 


1.18 ±0.01 


1.16 


0.38 


1.5 








1.020 ±0.010 


1.00 


0.33 


1.3 


0.42 % 



t (S52). t (S54, M10). 

direction, E' ntm - m , which corresponds to the forward cone just filling the 
forward hemisphere. The properties of the most important of these 
neutron sources follow: 

H 3 (p,n)He 3 

McKibben's nomograph for this reaction is shown in Fig. 31. Since 
tritium has only recently become extensively available, this reaction 
has not been very widely used for a neutron source; only the Wisconsin 
group has reported extensive (cross section) measurements with this 
source (B41, M31). Most of the information concerning yields, angular 
distributions, etc., vs. proton energies up to 2.5 Mev is due to the Los 
Alamos group (J2, H34). The cross section is shown in Fig. 32. The 
rapid increase may be due to a resonance of He 4 corresponding to in- 
cident protons of >2.5 Mev energy. The angular distribution of the 
emerging neutrons is quite complex (J2). A large yield of 20-Mev 



T(p,re)He 3 
3.0 



180^- 




E n (Mev) 

Fig. 31. McKibben nomograph of energy relationships in the H 3 (p,n) He 3 reaction 
(H34). Directions for its use are given on p. 375. 































































'o 




















e 
J 30 
















*? 
















































0.10 
















































1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 

•Ep(Mev) 

Fig. 32. Total cross section of the H 3 (p,n)He 3 reaction as a function of the incident 

proton energy {32). 
377 



378 



The Neutron 



[Pt. VII 




4.0 3,5 3.0 2.5 



,iiiii — r 

1.0 0.5 



2.0 E p 1.88 (Mev) 2.0 



3.0 3.5 4.0 



! I I I i I I M I I | i I I ' I" 
0.5 1.0 1.5 2.0 



E n (MeW 



Fig. 33. McKibben nomograph for the Li 7 (p,ra)Be 7 reaction (H34). Directions for 
its use are given on p. 375. 



11 










































































10 


- 








I 




























































/ 














\ F-40-kev Li target 










4 


' 


































J 


■ 


































/ 


■ 


































. 


' 








7 


























''E n (0°) 






























' 




































' 


















\ 
















. 


/ 




















\ 
















/ 




































/ 




















4 




1 












/ 
























\ 




j 






/ 




























\ 




/ 




[/_ 






























} 




J 


/ 




































1/ 




































































1 




} 


f 


































/ 









































































2.0 



s 

1-5 X 



1.0- 



0.5 



1.8 2.0 2.2 2.4 2.G 2.8 3.0 3.2 3.4 3.6 
Proton energy (Mev) 

Fig. 34. Yield of the Li 7 (p,«)Be 7 reaction in the forward direction (H3-4). 



Sec. 3B] 



Sources and Detectors 



379 



gamma-radiation, arising from the H 3 (p,7)He 4 reaction, accompanies 
the (p,n) reaction, and may interfere with some forms of neutron detec- 
tion (A30). 

Li 7 ( P ,n)Be 7 

This reaction has provided the most extensively used source of inter- 
mediate-energy neutrons. Its nomograph is shown in Fig. 33. The 



S 0.30 



0.20 







































L 


i 7 (p, 


«)B 


3 7 cross section as a function of 


energry' 




- 






2 Points from Be 7 activity 

^1 Calibration points from Mn 55 bath standardizatio 

o Points from integrated angular distributions 
















i 


Foir 


its I 


rom 


E1SS1C 


n ra 


tes 


































































t\ 




























/ : 


: 


\ c 

Y 


: 
































\ c 


1 




















































A 


i 
















J ( 








t 


. ' 


■> ■* 
























































J 




— ~»— — Extrapolation to threshold assumed linear in E„ 
£n£±£* Extrapolation to threshold assumed linear vas/E^ 






i 
i 































1.9 2.0 



2.1 2.2 2.3 

Proton energy (Mev) 



2.4 



Fig. 35. Total cross section for the Li 7 (p,ra)Be 7 reaction (H34). 

yield in the forward direction is shown in Fig. 34. The first peak is due 
to the concentration of neutrons in the forward cone for proton energies 
close to threshold. The second peak, at E p « 2.3 Mev, is real, and 
corresponds to a resonance of the compound nucleus Be 8 . The angular 
distribution is peaked in the forward direction, especially in the vicinity 
of the resonance, and requires terms up to cos 2 (P 2 ) for its description 
(Tl). The total cross section vs. proton energy is shown in Fig. 35. 



380 The Neutron [Pt. VII 

Until relatively recently, it was thought that the neutrons from this 
reaction were truly monoenergetic (H8). However, the reaction is now 
known to yield a low-energy group of neutrons corresponding to an ex- 
cited state of Be 7 at 435 kev. The results of various investigators on the 
relative yield of the low-energy group are summarized in Table 20. 





TABLE 20 




lTivb Yield of 


Neutrons Arising from Excitation of the 435-] 


State 


OF Be 7 IN THE 


Li 7 (p,m)Be 7 * Reaction 
Intensity Relative 




Proton Energy 


Angle of 


to Ground-State 




(Mev) 


Observation 


Group (%) 


References 


2.378 (E t *) 


0° 


~3 


W33 


2.705 


0° 


8±2 


H20 


2.75 


30° 


9± 1.5 


J5 


2.89 


30° 


10.5 ± 1 


J5 


3.120 


0° 


8±2 


H20 


3.31 


0° 


17 ± 10 


J8 


3.49 


0° 


10 ±3 


F42 


3.66 


30° 


12 ± 1 


J5 


3.91 


0° 


9±4 


J8 




60° 


16 ±6 


J8 



Additional Be 7 levels have been reported below 1 Mev (G30), but these 
have not been confirmed in other investigations (K6). 

Other Possibilities 

Other possible (p,n) reactions are discussed by Hanson, Taschek, and 
Williams (H34) and by Richards, Smith, and Browne (R10). The 
Be 9 (p,n)B 9 reaction is quite similar to that on Li 7 , but does not give a 
low-energy group up to neutron energies of ~1.5 Mev (J9). However, 
its yield is lower, and thin beryllium targets are considerably more 
difficult to prepare. 

The (p,n) reactions on scandium, vanadium, and manganese are 
being considered for producing monoenergetic neutrons in the ~l-kev 
energy range. Their yields have been studied by a number of investiga- 
tors (H34, R10, B4, M10, S52, S54). 

(6) Fast and Very Fast Neutrons from (d,ri) Reactions. These reac- 
tions are, with the exception of C 12 (d,n)N 13 , exoergic. They are useful 
in producing fast neutrons in the range 1-20 Mev. The Q values for the 
most useful (d,n) reactions are shown in Table 21. The last two yield 
heteroenergetic neutrons, but have been extensively used because of 
their high yields. 



Sec. 3B] 



Sources and Detectors 



381 



TABLE 21 
Properties op Some (d,n) Reactions 



Reaction Q Value (Mev) 

H 2 (d,re)He 3 3.265 ±0.018 (T16) 

H 3 (rf,w)He 4 17.6 (H34) 

C 12 (d,w)N 13 -0.281 ± 0.003 (T16) 

N 14 (d,n)0 15 5.1 (H34) 

Li 7 (d,n)Be s 15.0 (T16) 

Be 9 (d,n)B 10 3.79 (T16) 



E n0 for 

E d = (Mev) 

2.45 

14.1 

(E, = 0.328) 

4.8 

13.3 

3.44 



The last column gives the energy E n0 of neutrons which would result 
from the absorption of zero energy deuterons; this energy is independent 




1 2 3 4 5 6 7 

Energy of accelerated particle (Mev) 

Fig. 36. Energies of neutrons, emitted at 0° and 180°, for the most important mono- 
energetic (d,n) and (p,ri) reactions (H34). 

of the angle of emergence of the neutron. For deuterons with a given 
kinetic energy, the neutron energy is uniquely determined by its angle 
of emergence, provided that the reaction goes to the ground state of the 
product nucleus. The neutron energies vary on both sides of E n0 , being 
greater for emission in the forward direction and less for backward 
emission. The energies of neutrons emitted at 0° and 180° are shown in 
Fig. 36, as a function of the bombarding energy, for the first three reac- 



382 The Neutron [Pt. VII 

tions in the table. Also included are curves for the two (p,ri) reactions 
discussed above. 

H 2 (d,n)He 3 

The d-d reaction has been extensively used, and its characteristics 
widely investigated (H34). It has high yields for low bombarding 
energies, which makes it particularly useful as a neutron source with 
low-voltage (<1 Mev) particle accelerators. The neutrons are mono- 
energetic up to bombarding energies of 10 Mev. Figure 37 is the Mc- 
Kibben nomograph for this reaction. 




EV(Mev) 2 3 



Fig. 37. McKibben nomograph for the d-d reaction (H34). Directions for its use 

are given on p. 375. 

The angular distribution of d-d neutrons has been extensively studied, 
both experimentally and theoretically. At energies of a few hundred 
kev, the distribution in the cm. system is symmetrical about 90°, being 
~60 percent lower at 90° than at 0° and 180° (H34, M16). For higher 
energies the distribution becomes much more anisotropic. Hunter and 
Richards (H78) have investigated the angular distribution for deuterons 
of from 0.5 to 3.7 Mev. They find that it can be represented by an ex- 
pression of the form 

N(fi) = A P (d) + A 2 P 2 {6) + AiP^e) + A 6 P 6 (d) +■■■ (87) 

in which the P„(0)'s are the Legendre polynomials, and all the coeffi- 
cients vary with energy, as shown in Fig. 38a. The theory of the 
angular distribution has been discussed by Konopinski and Teller (K21) 



Sec. 3B] 



Sources and Detectors 



383 



and by Beiduk, Pruett, and Konopinski (B14). The total cross section 
vs. deuteron energy is shown in Fig. 38b. 




E d (Mev) 



Fig. 38a. Variation with deuteron energy, E d , of the coefficients of the Legendre 

polynomial fit to the angular distribution of the neutrons from the d-d reaction in 

the center-of-mass coordinate system (H78). 





' 


' ' 


i I r 




0.10 


o 


^S^~* " "° ' 


"^ s— * 


• 


^0.08 


D / 


Symbol on 


Reference No. 




01 


o 


Figure 


in Legend 




S, 0.06 
to 


- o* . 


• 

Q 


1 

2 
3 




0.04 


- & 

.8 


A 


4 
5 




0.02 


a 
- + 









E d (Mev) 

(1) Hunter and Richards (H78). (2) Blair, Freier, Lampi, Sleator, and Williams, Phys. Rev., 74, 
1599 (1948). (3) Manley, Coon, and Graves, Phys. Rev., 70, 101(A) (1946). (4) Graves, Graves, 
Coon, and Manley, Phys. Rev., 70, 101(A) (1946). (5) Bretscher, French, and Seidl, Phys. Rev., 

73, 815 (1948). 

Fig. 38b. Total cross section for the H 2 (d,n)He 3 reaction (H78). 

The d-d reaction is frequently used to provide strong neutron sources 
by bombarding thick targets of heavy ice with deuterons. The thick 
target yield curve (H34) is given in Fig. 39. 



384 



The Neutron 



[Pt. VII 



The companion reaction H 2 (d,p)K 3 has a comparable cross section 
(M16, H34). The protons are frequently used to "monitor" the neutron 
yield. At high bombarding deuteron energies the He 3 recoils from the 






1 

/2 



10 'o 



2 5 



0.2 



0.4 0.6 0.8 

Deuteron energy (Mev) 



1.0 



Fig. 39. Thick heavy ice target H 2 (d,ra)He 3 neutron yields (H34). 

(d,n) reaction can be used as a monitor, and also, by observing He 3 
neutron coincidences, to eliminate background effects due to (d,ri) re- 
actions on target impurities. 

H 3 (d,w)He 4 

This, the t-d reaction, is the most strongly exoergic of the reactions 
capable of yielding monoenergetic neutrons; it can provide neutrons of 
from 12 to 20 Mev by using deuterons of up to 3 Mev (Fig. 36). The 
McKibben nomograph for the reaction is shown in Fig. 40. This reac- 
tion is especially useful with low-voltage deuteron accelerators, since it 
has very large yields for deuterons of a few hundred kev energy, owing to 
a resonance in the cross section at ~100 kev. The low-energy cross- 
section data (up to ~1 Mev deuterons) can be fitted by the expres- 
sion (H34) 

58 exp (-1.72/2^) 
a = — — — — — barns (88) 



E 1 + {E - 0.096)7(0. 174)' 



where E is the deuteron energy in Mev. From 1 to 2.5 Mev the cross 
section remains essentially flat at ~0.15 barn. The cross section is 0.05 
barn at 10.5 Mev (B72). 



Sec. 3B] 



Sources and Detectors 



385 



The angular distribution in the t-d reaction, for incident deuterons up 
to 2.5 Mev, has been measured by Taschek, Hemmendinger, and Jarvis 
(T2) by observing the distribution of recoil alpha-particles. Their re- 



T(d,m)He 4 




10 20 



E {Mev) 10 



Pig. 40. MeKibben nomograph for the t-d reaction (H34). Directions for its use 

are given on p. 375. 

suits are shown in Fig. 41. Detection of the recoil alphas provides an 
effective means for monitoring the neutron yield from this source. The 
angular distribution for E d = 10.5 Mev has been measured by Brolley, 
Fowler, and Stovall (B72). 



~0.016 

"3 

* 0.012 



"0.008 



^-^- T 




fflEJ-- 




2.5 | ^4-A°;f<4.2.5 


1.5 

— - ± 



! 0.024 



40 50 60 70 80 90 100 110 120 130 
Laboratory angle (degress) 




30 60 90 120 150 

Center of Gravity angle (degrees) 



Fig. 41. Angular distribution of recoil alpha-particles from the H 3 (<2,n)He 4 reac- 
tion for various deuteron energies (in Mev) (T2, H33). 



386 The Neutron [Pt. VII 

Because of the strong low-energy resonance, the t-d reaction is expected 
to have large thick target yields. Thus, for a thick gas target and 600- 
kev d's the yield is 5 X 10 8 neutrons per microcoulomb ; for tritons ad- 
sorbed in a thick zirconium target, the yield at Ed = 200 kev is about 
10 8 neutrons per microcoulomb (H34). 

One interesting application of the t-d reaction is in the possibility of 
converting intense sources of thermal neutrons (such as those available 
from piles) into strong fast-neutron sources. Thus, if thermal neutrons 
are allowed to fall on a mixture of lithium and deuterium, fast tritons 
(E = 2.65 Mev) will be produced by the Li 6 (n,a)H 3 reaction. These 
tritons then react with the deuterons to produce 14-Mev neutrons by 
the t-d reaction. A recent investigation by Almqvist (All) has shown 
that such sources also produce neutrons by reactions of tritons on the 
lithium nuclei; the latter reactions have a considerably greater yield 
than the t-d reaction in the mixtures used. Since the neutrons from 
lithium result from a variety of reactions on both lithium isotopes, with 
Q values ranging from 16 to 8.9 Mev, the neutrons from such sources 
have a complex spectrum. The neutron yields for a number of com- 
pounds are shown in Table 22. From these data the author derives 

TABLE 22 

Neutron Yields fkom Thermal Neutron Irradiation of Lithium and 
Lithium-Deuterium Compounds (All) 

Neutrons per 10 6 Tritons 



Substance 


From Li 


From '. 


LiF 


26.8 




Li 2 C0 3 


17.2 




LiOH-H 2 


15.2 




LiOD 


20.3 


3.4 


LiODD 2 + D 2 


9.3 


12.1 



average cross sections of 1.5 and 0.41 barns for 2.65-Mev tritons on 
thick targets of lithium and deuterium, respectively. Crews (C32) has 
measured the yield and angular distribution of neutrons from thin 
lithium targets for triton energies between 0.25 and 2.10 Mev; he 
derives a ~ 0.76 barn for E t = 2.0 Mev on a thick Li target. 

C 12 (rf,n)N 13 

This, among the (d,n) reactions of interest, is the only endoergic 
reaction. It is mainly useful because of its low threshold (Table 21; 



Sec. 3B] Sources and Detectors 387 

Fig. 36), which makes it a convenient low-voltage accelerator source for 
neutrons of from a few hundred kev to ~1.5 Mev. Since the first ex- 
cited state of N 13 , observed in this reaction, is at ~2.3 Mev (G29), the 
neutrons are monoenergetic to —2 Mev. The reaction yield is rela- 
tively low and shows a complex angular distribution and energy de- 
pendence (H34). 

The companion reaction C 12 (d,p)C 13 has a similar yield curve at low 
deuteron energies, and is useful for monitoring the neutron yield. A 
complicating feature is the presence, with normal carbon targets, of the 
reaction C 13 (d,n)N 14 , with a Q value of 5.2 Mev, which gives a fast- 
neutron group of intensity <~1 per cent of the C 12 neutrons. 

N 14 (<Z,n)0 15 

This reaction, very seldom used, gives monoenergetic neutrons for 
deuterons of up to ~1 Mev energy. At 1 Mev the cross section relative 
to that of the d-d reaction has been measured by Gibson and Livesey 
(G4), and is shown in Table 23. 

TABLE 23 

Ratio of d-d to N 14 -rf Cross Sections for 1-Mev Deuterons (G4) 

Angle Ratio (d-d/N u -d) 
0° 4.4 

30° 4.4 

90° 1.5 

150° 2.1 

U 7 (d,n)Be 8 

Because of its large Q value and large yield, this source has been exten- 
sively used, especially with cyclotrons. However, the large number of 
levels, in both the compound and product nuclei, gives rise to a com- 
plicated neutron spectrum with a wide energy spread (H34). The thick 
target yield for a bombarding energy of 600 kev is 17 X 10 6 neutrons 
per microcoulomb. Above E d = 1 Mev the thick target yields are 
greater than for the d-d reaction. 

Be 9 (d,w)B 10 

This reaction also gives rise to a complex neutron spectrum of lower 
maximum energy than that from Li 7 (A4). The thick target yields at 



388 



The Neutron 



[Pt. VII 



low deuteron energies are shown in Table 24. Above ~1 Mev the thick 
target yields are between those of the d-d and the Li 7 (d,ri) reactions. 

TABLE 24 
Thick Taeget Yields fob the Be 9 (d,w)B 10 Reaction 
Ed (kev) 10 6 Neutrons/Microcoulomb 



400 


4 


600 


21 


800 


106 



Other Possibilities 

Many other (d,n) reactions have been employed as neutron sources. 
Among these the F 19 (d,ra)Ne 20 , Q = 10.7 Mev, has received some atten- 
tion, but its characteristics are not more favorable than the other reac- 
tions which yield heteroenergetic sources, and its yield is lower. While 
such sources have frequently been used in the past to produce very fast 
neutrons, they have been mainly superseded by the t-d reaction. 

The (d,ri) yields of a variety of elements, for deuteron energies of 
~10 Mev and greater, have been studied by a number of investigators. 
Thick target yields for 10-Mev deuterons (due to Smith and Kruger, 
S41) are shown in Table 25, together with the yields at 15 Mev (due to 
Allen, Nechaj, Sun, and Jennings, A8). The latter results are sur- 

TABLE 25 
Neutron Yields from 10- and 15-Mev Deuterons on Thick Targets 







Yield (10 8 neutrons/ 






Yield (10 s neutrons/ 






microcoulomb) 






microcoulomb) 


Target 


Z 




Target 


Z 






10 Mev 


15 Mev 


10 Mev 


15 Mev 


Be 


4 


320 


190 


Nb 


41 




15 


B 


5 


190 




Mo 


42 


42 


15 


C 


6 


120 




Ag 


47 




14 


Al 


13 


87 


64 


Cd 


48 




12 


P 


15 


105 




Sb 


51 


35 




Ti 


22 




65 


Ta 


73 


7.4 


3.3 


Cr 


24 




29 


W 


74 


7.0 




Mn 


25 


76 


52 


Pt 


78 


6.0 




Co 


27 




26 


Au 


79 


4.7 


2.1 


Ni 


28 


33 




Pb 


82 




2.1 


Cu 


29 


55 


29 


Bi 


83 




1.3 



Sec. 3B] Sources and Detectors 389 

prisingly lower than those at 10 Mev, although the trend with atomic 
number is the same for both series of measurements. The discrepancy 
probably arises, at least in part, from the fact that the 10-Mev measure- 
ments counted neutrons of all energies while the 15-Mev results apply 
only to those neutrons capable of exciting the S 32 (n,p) reaction. 

The angular distributions of (d,n) neutrons have been investigated by 
Roberts and Abelson (R14), by Falk, Creutz, and Seitz (Fl), by Allen 
et al. (A8, S58) (all at E d = 15 Mev), by Ammiraju (A22) {E d = 18 
Mev), by Schecter (S4) {E d = 20 Mev), and by others. These distri- 
butions show a strong peaking in the forward direction. [Some, e.g., 
Be, when only the highest-energy neutrons are detected, show structure 
in the angular distribution (S58) .] 

The observed characteristics of the (d,ri) reaction for high-energy 
deuterons can be explained in terms of the "stripping" theory of Serber 
(S20) and Peaslee (P8), according to which most of the neutrons are 
produced by a process in which the neutron never enters a compound 
nucleus; the deuteron is polarized in the field of the nucleus and then 
split, the proton being captured (and occasionally scattered) by the tar- 
get nucleus. Recently, Butler (B78, B79) has made a significant con- 
tribution to the "stripping" theory by considering in greater detail the 
angular distribution of neutrons from such reactions when the product 
nucleus is left in a definite quantum state. His calculations predict 
structure (i.e., maxima and minima) in the angular distribution which 
can be interpreted in terms of the angular momentum and parity 
properties of the initial and final states. 

(c) Ultrafast Neutrons from Accelerated Deuterons and Protons. When 
charged particles from ultrahigh-energy accelerators strike a target, 
neutrons are produced through a variety of reactions. Those reactions 
which involve the capture of the bombarding particle into a compound 
nucleus give rise to neutrons by the process of evaporation. Such neu- 
trons have the broad energy distribution characteristic of the statistical 
theory, with the maximum in the fast- or very fast-energy region and a 
more-or-less symmetrical angular distribution. However, when deu- 
terons (and, to a lesser extent, protons) are used as the bombarding 
particles, there is observed, superimposed on the evaporated neutrons, 
a strong forward peak of ultrafast neutrons with a relatively narrow 
energy distribution. In the case of deuteron bombardment, the neu- 
trons arise mainly from the stripping process, mentioned briefly in the 
preceding section. Serber (S20) has given a simple picture of this 
process, as follows: 

The deuteron is a relatively loosely bound structure in which neutron 
and proton spend a large fraction of the time far apart. Hence, it is 



390 The Neutron [Pt. VII 

not improbable that, as a deuteron traverses the target, the proton will 
strike one of the target nuclei while the neutron remains outside the 
nucleus. In such a collision the proton will be stopped while the neu- 
tron will go on, carrying off approximately half of the original deuteron 
energy. 

However, the neutron has internal motion, relative to the center of 
mass of the deuteron. Hence, as the neutron leaves the proton behind, 
this relative motion is superimposed on the forward motion of the 
deuteron, giving rise to an energy spread centered around the average 
value of half the deuteron energy. Another result of the relative motion 
is that the emergent "stripped" neutrons have a spread in directions, 
around the original deuteron direction. 

The magnitudes of the energy and angular spreads of the neutron 
beam can be estimated as follows: The forward momentum of the 
neutron, due to the kinetic energy, E d , of the deuteron, is 

Po = (ME d ) 1A (89a) 

The internal momentum of the neutron is, roughly, 

Vi ^ (Me d ) y ° (89b) 

where t d is the deuteron binding energy. The stripped neutrons have 
energies within the limits given by 

(po ± Pi) 2 1 
E ^ n P ~2 Ed± {tdEd) (90) 

A more accurate calculation (S20) gives, for the full width at half 
maximum, 

AE = 1.5(tdE d ) H (90a) 

The angular spread in the neutron beam is determined by the relative 
values of the forward and transverse neutron momenta: 

A0 « =* 2—1 radians (91) 

Po \eJ 

The more accurate calculation (S20) gives 

A0 = 1.6 (|-) ' (91a) 

as the full angular width at half maximum. The above considerations, 
with respect to the angular spread, apply only to targets of light nuclei, 
in which the coulomb deflection of the deuterons is negligible. For the 



Sec. 3B] 



Sources and Detectors 



391 



heaviest nuclei the coulomb deflections lead to about twice as large a 
spread as Eq. (91a). 

The energy and angular distributions of the stripped neutrons are 
superimposed on the background of neutrons produced as a result of 
evaporations. However, this background is expected to be small, 
especially since it is spread, approximately uniformly, over all direc- 
tions. In addition, some neutrons are produced by the disintegration 
of deuterons in the coulomb field of the target nuclei; this effect is rela- 
tively small for light nuclei and becomes comparable to the stripping 
production for the heaviest target materials (Dl, K20). 




40 60 80 100 

Neutron energy (Mev) 



120 



140 



Pig. 42. Energy distribution of neutrons obtained by stripping 190-Mev deuterons 

in a 1.27-cm thick beryllium target, due to Hadley et al. (HI). The curve is from 

the theory of Serber (S20). 

The observations of Helmholz, McMillan, and Sewell (H53) on 
angular distributions of the neutrons produced by bombarding various 
nuclei with 190-Mev deuterons and of Hadley, Kelly, Leith, Segre, 
Wiegand, and York (HI) on the energy distribution of the neutrons 
from a beryllium target bombarded by 190-Mev deuterons are in good 
agreement with Serber's theory, and demonstrate the possibility of 
using the stripping process to obtain relatively monoenergetic ultra- 
fast neutron beams from ultrahigh-energy particle accelerators. The 
energy distribution due to the latter investigators is shown in Fig. 42. 

When ultrafast protons are used as the bombarding particles, the 
emerging neutrons are neither so sharply collimated nor so nearly mono- 
energetic. However, observations by Miller, Sewell, and Wright (M32) 
(330-Mev protons on beryllium, aluminum, copper, and uranium) and 
by DeJuren (Dll) (340-Mev protons on beryllium) have demonstrated 
that there is an appreciable component of ultrafast neutrons emitted at 
forward angles. The results of the first-mentioned investigators, on the 
full angular widths at half maximum for neutrons of E > 20 Mev, are: 
beryllium, 54°; aluminum, 59°; copper, 59°; uranium, 58°. 



392 The Neutron [Pt. VII 

Such forward-collimated, ultrafast neutrons result primarily from one 
or a few p-n collisions in the nucleus ; at these energies the scattering of 
neutrons by protons is predominantly forward (charge exchange), and 
the scattered neutron has an appreciable probability of emerging from 
the nucleus without undergoing any further scattering. 

Relative neutron yields for targets of various Z have been measured 
by Knox (K20) with both high-energy deuterons and protons as the 
bombarding particles. For 190-Mev deuteron bombardment the rela- 
tive yields agree well with stripping plus disintegration by the Coulomb 
field. In the case of 340-Mev proton bombardment, the yield varies 
approximately as (A — Zf 3 . 

The energy distribution of the neutrons from beryllium and carbon 
targets bombarded with ~100-Mev protons has been investigated by 
Bodansky and Ramsey (B42, B43). For a beryllium target they ob- 
serve a sharp peak at E n « 93 Mev, with a width at half maximum of 
~30 Mev. Below ~70 Mev, the neutron energy distribution becomes 
essentially fiat, with a yield of ~40 percent of the peak value. The neu- 
tron yield from a carbon target shows a slight peak at ~70 Mev (~25 
percent above the roughly constant yield below E n = 60 Mev) and 
falls off rapidly above 70 Mev. Similar studies on neutrons from beryl- 
lium, carbon, aluminum, and uranium bombarded with 170-Mev pro- 
tons have been made by Taylor, Pickavance, Cassels, and Randle (T3). 

4. Neutrons from U 235 Fission. One of the most important present- 
day neutron sources is the fission reaction. Fission of the heaviest ele- 
ments is accompanied by the emission of fast neutrons. The neutron 
spectrum resulting from thermal neutron fission of U 235 is closely 
approximated by the expression (U2) 

<W 

— • = e-^sinh(2£/) M (92) 

dE 

where E is the neutron energy in Mev. This expression, suggested by 
Watt, agrees with the observed fission spectrum to within ±15 per- 
cent up to E = 17 Mev. 1 Equation (92) is plotted in Fig. 43. 

As seen in the figure, the fission spectrum has a rather broad energy 
distribution, with an average energy of ~1.5 Mev. However, this is by 
no means the spectrum of neutrons normally observed either inside or 
emerging from piles, since neutrons in piles suffer considerable modera- 
tion by elastic scattering on light nuclei or by inelastic scattering on 
heavy nuclei. Nevertheless, strong sources of unmoderated fission neu- 

1 See T. W. Bonner, R. A. Ferrell, and M. C. Rinehart, Phys. Rev., 87, 1032 
(1952); D. L. Hill, Phys. Rev., 87, 1034 (1952); B. E. Watt, Phys. Rev., 87, 1037 

(1952). 



Sec. 3B] 



Sources and Detectors 



393 



trons can be obtained by irradiating a uranium target with an intense 
thermal neutron beam (as from a pile). Such a source was used in the 
measurements by Hughes et al. (H72) of capture cross sections for fast 
neutrons, discussed in Section 2. 



0.9 
0.8 
0.7 
0.6 
0.5 
0.4 
0.3 


















































































































































































































0.1 























































1 2 3 4 5 6 7 8 9 10 11 12 13 

E n (Mev) 

Fig. 43. Approximate shape (to within ±15%) of the neutron spectrum from the 
fission of U 236 by thermal neutrons (U2). 

5. Neutrons in the Cosmic Radiation. Because the neutron is un- 
stable, it is not expected to be one of the components of the primary 
cosmic radiation. Nevertheless, neutrons constitute an appreciable frac- 
tion of the "hard" or "penetrating" component of cosmic rays. These 
neutrons arise from nuclear disintegrations (stars) in the atmosphere 
and are responsible, in turn, for a large fraction of the observed nuclear 
disintegrations, especially at low altitudes (R21, S38, K22). 

The observations on neutrons in the cosmic radiation are of two 
general types: 

(1) The detection of very fast and ultrafast neutrons by observation 
of their interaction with nuclei, especially star production. Of such 
experiments, the most unambiguous are those in which stars are observed 
in sensitive nuclear emulsions, where the particle responsible for the 
disintegration can be seen or, in the case of neutrons, deduced from the 
absence of any charged particle of sufficient energy. Results of such 
experiments are reported and summarized by Bernardini, Cortini, and 
Manfredini (B18). 



394 The Neutron [Pt. VII 

(2) The detection of slow neutrons in paraffin-surrounded and 
cadmium-covered BF 3 counters (Y4, S50). Many such experiments 
have been performed by many investigators; one of the most recent and 
most extensive, due to Yuan (Y4), has been carried to an altitude of 
~100,000 ft (pressure ~1 cm Hg). The thermal neutron intensity 
shows an exponential increase (mean free path ~ 156 g/cm 2 ) up to 
about 20 cm of Hg, a maximum at ~8.5 cm of Hg, and a rapid falling 
off at higher altitudes. 

The interpretation of experiments of the second type requires con- 
sideration of the processes by which the neutrons, produced mainly as 
fast neutrons in nuclear disintegrations, lose energy by scattering and, 
at the same time, diffuse through the atmosphere. (The general dis- 
cussion of such problems is reserved for Section 4.) The theory of the 
slowing down of neutrons in the atmosphere has been given by Bethe, 
Korff , and Placzek (B26) ; the diffusion problem is discussed by Flugge 
(F35). Their calculations have recently been revised (in the light of 
more accurate data on nuclear cross sections) by Davis (D5) and by 
Lattimore (L6). 

6. Sources of Slow Neutrons. By far the greatest number of neutron 
investigations have involved the use of slow neutrons. Sources of slow 
neutrons are invariably derived from intermediate- or higher-energy 
neutrons by allowing the neutrons to diffuse through a "moderating" 
material. [The lower limit of monoenergetic neutrons from threshold 
charged particle reactions, mostly (p,n), is being slowly pushed toward 
the slow-neutron region.] 

Neutrons lose energy through inelastic collisions with medium or 
heavy nuclei and through elastic collisions with light nuclei; the first 
process is most effective for fast neutrons and the second for interme- 
diate and slow neutrons. Consider a source of fast neutrons embedded 
in a large mass of material, preferably one containing light nuclei. The 
neutrons diffuse through the material, losing energy as they suffer scat- 
terings. If the capture cross section is small compared to the scattering 
cross section (as it is in the intermediate- and fast-neutron ranges for 
most materials, especially light elements), the neutrons continue to lose 
energy until their energy is comparable to the energy of thermal agita- 
tion of the nuclei in the moderating material. After this, a neutron is 
as likely to gain as to lose energy in a scattering, and the slowing down 
stops. The neutrons are then said to be in "thermal equilibrium" with 
the moderating medium. 

The energy distribution of neutrons in thermal equilibrium is in many 
respects similar to that of the molecules of a gas at the temperature of 



Sec. 3B] Sources and Detectors 395 

the medium. It is closely approximated by the Maxwell distribution 
function, 

dn 

— = Av 2 e- M * 2 /M T (93) 

dv 

where M is the neutron mass, k is Boltzmann's constant, and T is the 
absolute temperature of the moderating medium; the factor A is gen- 
erally a function of the position in the medium. The Maxwell distribu- 
tion gives as the most probable velocity 



o'-i^) (94) 



/2kT\ 
\W 

which is 2.2 X 10 5 cm/sec for T = 300°K; the average velocity is 

2 

v = —v' = 1.128?/ (95) 

Strictly speaking, in an infinite homogeneous moderating medium the 
thermal neutrons will have a Maxwell energy distribution only under 
special circumstances, of which two are: (1) the nuclei in the moderating 
medium scatter but do not absorb neutrons; (2) the neutrons are emitted 
into the medium with the Maxwell energy distribution and the absorp- 
tion cross section obeys the 1/v law. In this case, since the rate of 
neutron absorption, ~nw(v), is proportional to n, the original distribu- 
tion shape will maintain itself. 

Most thermal neutron sources deviate somewhat from the Maxwell 
distribution. If the medium strongly absorbs neutrons in the thermal 
energy range, the neutrons will never attain very low velocities, and the 
average velocity will be greater than that given by Eq. (95). If the 
size of the slowing-down medium is not sufficiently great, the neutrons 
will escape before attaining thermal equilibrium. In this situation, the 
average energy will again be greater than in Eq. (95). In any event, 
there is always present in the medium, at distances relatively close to 
the source, a component of neutrons in the process of being slowed down. 
These have an energy distribution characterized by equal fluxes (<£ = nv) 
of neutrons in equal logarithmic energy intervals, i.e., 

BdE 
d<j> = — (96) 

where B depends both on the position in the medium and on the neutron 
energy. Such neutrons (called epithermal) are observed as a tail on the 
Maxwell distribution. 



396 The Neutron [Pt. VII 

The theory of slowing down and diffusion of neutrons will be discussed 
in Section 4. Some results, useful in the planning and interpretation of 
experiments involving slow-neutron sources, are summarized below. 

The most frequently used moderating media are water or paraffin, and 
graphite. The former have the advantage of containing a large propor- 
tion of hydrogen, the most effective nuclei for slowing down intermediate 
energy neutron; hence, for a given source strength, higher thermal 
neutron densities can be obtained with smaller quantities of moderating 
material than for any other slowing-down medium. On the other hand, 
carbon has a considerably smaller capture cross section than hydrogen, 
so that it is possible to obtain purer thermal neutron sources by making 
observations at greater distances from the fast-neutron source. Other 
materials which have proved useful as slowing-down media include 
heavy water, beryllium, and beryllium oxide (BeO); these are inter- 
mediate between water and graphite in slowing-down effectiveness, but 
are considerably more difficult to obtain in sufficient quantity. 

Since the early work of Fermi and co-workers (F17, A20) on the 
moderating properties of water there have been many investigations of 
slow-neutron distributions in water for many fast-neutron sources. The 
results of one such study, due to Anderson, Koontz, and Roberts, are 
shown in Fig. 44, a plot of activity X r 2 (proportional to r 2 nv) vs. dis- 
tance from the source. The characteristic features of such curves are 
the peak at a relatively small distance and the exponential decrease at 
large distances from the source. 

The results of such studies are frequently summarized in terms of the 
value of the second moment of the distribution, 



| Ar 4 dr 
-* ° - (97) 






Ar 2 dr 



where A is the measured activity of the detector used. The measure- 
ments of Anderson, Koontz, and Roberts (Fig. 44) give r 2 = 278 and 
353 cm 2 for the indium resonance (1.44 ev) and thermal neutron detec- 
tor, respectively. (These are in fair agreement with the results of other 
investigators.) The characteristic lengths of the exponential decrease 
at large distances, 

r 2 A(r)~e- rl * (98) 

were found to be X = 9.43 and 10.00 cm for the indium resonance and 
thermal detectors, respectively. 



Sec. 3B] 



Sources and Detectors 



397 



The slowing down of neutrons in graphite has also been studied with 
various sources. The results of measurements by Feld and Fermi are 
shown in Fig. 45 as curves of thermal and indium resonance detector 



2.0 



1.6 



0.2 













Tr 




















( « 


K 












/ 




































/ 


j 


r\ 
































/ 


/ 








































































1 












ierrr 


al n 


BUtr 


ons 


x 1( 


-' 










11 










































/ 


































1 


/ 




































/ 










































































































1/ 

1 1 




































/ 1 
1 1 




































// 




































'/ 


































(1 


/ 


































l\ 


























Nj 










1 









































































12 16 20 24 

Distance from source (cm) 



28 



Fig. 44. Distribution of slow neutrons from a Ra-a-Be source in water. The ordi- 
nate is the activity X r 2 of a thin indium foil. The curve labeled "indium reso- 
nance" represents the activity of a cadmium-covered foil; the curve labeled "thermal 
neutrons" represents the activity of a bare foil minus 1.07 times the activity of the 
cadmium-covered foil. The source had a strength of 13.2 X 10 6 neutrons/sec. 
The indium foils had an area of 3 X 6.4 cm 2 and a thickness of 1.69 g/cm 2 . The 
cadmium covers were 0.96 g/cm 2 thick. Data due to Anderson, Koontz, and Roberts 
(unpublished). The ordinate scale can be converted to a scale of thermal neutron 
flux (hv) X r 1 by multiplying by 1.6 X 10 6 . 



activation vs. distance from the source, for a Ra-a-Be source (Fig. 45a) 
and a Ra-7-Be source (Fig. 45b). Since these measurements were made 
in a graphite column of non-negligible lateral dimensions, the geometry- 
is not strictly comparable to that for a point source in an effectively 
infinite medium (Fig. 44). However, the geometric effects can be taken 
into account in interpreting such experiments (see Section 4). 



398 



The Neutron 
Ra-a-Be 



[Pt. VII 







V 




























180 




\\ 






























\ 




























160 






V 


— Tl 


erm 


il 






















1 


\ 
































\\ 


























































120 






























































100 




































\ 


























80 






\ 
































\ 
































\ 
































\ 




•\ 






















I 
res 


idiur 
onar 


n — - 
ce 


" > 








































• 




































































s 













































120 



Z (cm) 



Fig. 45a. Thermal neutron flux and indium resonance neutron flux for a Ra-a-Be 
source in graphite, density 1.6 g/cm 3 , due to Feld and Fermi (unpublished). All 
the data are normalized to the same value at z = 10.2 cm. The neutrons were 
slowed down in a rectangular graphite column, 4 ft by 4 ft square by 7 ft high. The 
source was placed on the long axis 2 ft from the bottom face of the column. The 
abscissa z is distance, along the column axis, between source and indium detector. 
The actual ratio of thermal to indium resonance activity at z = 10.2 cm for the 

detector used was 3.72. 

For a monoenergetic source in a graphite column, the distribution of 
indium resonance neutrons is expected to be Gaussian: 

Wm» = ae~ ( * /zo)2 (99) 

The distribution of indium resonance neutrons for the Ra-a-Be and 
Ra-Y-Be sources could be expressed as a superposition of 3 and 2 Gaus- 
sian functions, respectively, with the proportions and ranges given in 
Table 26. 



Sec. 3B] 



200 



Sources and Detectors 
Ra-y-Be 



399 



80 



20 





K 




























\ 


























































\ 




























V 


-Th 


ermj 


ii 






















\ 




























\ 




























I 


\ 




























\ 




























\ 




























\ 


















































































' 


































^ 










































• 




Indium 
resonance 













































































































60 80 

Z (cm) 



100 



140 



Fig. 45b. Thermal neutron flux and indium resonance neutron flux for a Ra-7-Be 
source in a graphite column (see legend of Fig. 45a), due to Feld and Fermi (unpub- 
lished). The actual ratio of thermal to indium resonance activity at 2 = 10.2 cm 
for the detector used was 2.57. The ratio of the strength of the Ra-a-Be source 
(see Fig. 45a) to the Ra-7-Be source was 0.231. The corresponding ratio of thermal 
neutron activities at 10.2 cm was 0.355. 



The distribution of thermal neutrons is a somewhat more complicated 
function of distance from the source. However, at distances greater 
than ~60 cm, for the Ra-a-Be source, and ~30 cm, for the Ra-y-Be 
source, the thermal neutron density becomes exponential : 



(nv) th = 0e-* /x 



(99a) 



with the same value of X = 24.2 cm applying to both sources. It is the 
difference between the exponential (Eq. 99a) and Gaussian (Eq. 99) 
functions which results, in graphite, in the rapid increase of the ratio of 



400 The Neutron [Pt. VII 

TABLE 26 

Analysis of Indium Resonance Neutbon Distributions along the Axis 

op a Graphite Column fob Ra-a-Be and Ra-7-Be Sources in Terms of 

Gaussian Functions (Eq. 99 and Fig. 45) 

Ra-a-Be Ra-7-Be 

«» (%) «o (cm) oci (%) z (cm) 

15.0 22.8 38.7 22 

69.3 36.9 61.3 29 

15.7 57.1 

meanf 39.2 meant 26.5 

t The mean is computed as the square root of the weighted mean square. 

thermal to epithermal neutron flux with increasing distance from the 
source. 

While slow-neutron sources of the types described above have been 
and continue to be extensively used, the most potent available slow- 
neutron sources are those associated with chain reactors (piles). The 
spectrum of neutrons inside a slow-neutron reactor contains an appre- 
ciable component of fast, intermediate, and epithermal neutrons. 1 How- 
ever, by use of a "thermal column," it is possible to obtain high fluxes 
of practically pure thermal neutrons. The usual thermal column is 
simply a large block of graphite placed directly adjacent to the pile. 
Neutrons which enter the column are slowed down as they diffuse to the 
outside, the ratio of epithermal to thermal neutrons falling off rapidly 
with the distance into the column. Figure 46 shows the activity of bare 
and cadmium-covered indium foils at various positions in the thermal 
column adjacent to the Argonne heavy-water-moderated pile. 

Thermal neutron fluxes, available from various sources, are summar- 
ized in Table 27. 

For many purposes, the thermal neutrons diffusing in a mass of 
moderating material can be utilized directly, in situ. Thus, for example, 
in irradiating a sample for the purpose of observing the resulting radio- 
activity, the maximum slow-neutron flux and consequently the maxi- 
mum activation are attained when the sample is embedded in the 
moderator as close as possible to the source. 

For some investigations, on the other hand, it is desirable to have a 
beam of thermal neutrons, e.g., for the measurement of cross sections by 
a transmission experiment. One method of obtaining a beam of thermal 
neutrons is to collimate (with a cadmium or boron slit system) a portion 
of the neutrons which are diffusing out through the bounding surface of 

1 For the spectrum of neutrons leaving a pile, see Fig. 54. 



Sec. 3B] 



Sources and Detectors 



401 



10,000 



1000 




0.001 

20 
71 cm->j 
from face 
of tank 



100 120 140 160 180 200 220 2 
Cm from face 



260 280 
-End of 
graphite 



Fig. 46. Activities of bare and cadmium-covered indium foils in the thermal column 
of the Argonne heavy water pile. The thermal column is of graphite, 5 ft square and 
240 cm long. Its front face is separated from the reactor by 71 cm of graphite. The 
average thermal neutron flux (nv) can be obtained by multiplying the ordinates 
corresponding to the bare indium foils by 0.10. The number of neutrons per cubic 
centimeter passing through the indium resonance energy per second (slowing-down 
density) can be obtained by multiplying the ordinates corresponding to the cadmium- 
covered indium foils by 0.0019. Measurements due to Seren (unpublished). 



402 



The Neutron 



[Pt. VII 



TABLE 27 
Some Available Thermal Neutbon Fluxes 



(a) Laboratory Sources 





Strength 




Thermal nv 


Source 


(n/sec) 


Position 


(n/cm 2 • sec) 


1 curie of Ra-a-Be in 


13.2 X 10« 


4 cm from the source 


8.5 X 10 4 


infinite H 2 




10 cm from the source 


2.9 X 10 4 






20 cm from the source 


0.31 X 10 4 






50 cm from the source 


0.0025 X 10 4 


1 curie of Ra-a-Be in 


14 X 10 6 


at source 


0.7 X 10 4 


graphite column, 




10 cm from the source, 


0.65 X 10 4 


4 ft x 4 ft x 7 ft; 




on the axis 




source 2 ft from 




50 cm from the source, 


0.25 X 10 4 


base, on axis 




on the axis 








100 cm from the source, 


0.04 X 10 4 






on the axis 




2 curies of Ra-y-Be 


3.2 X 10 6 


at source 


0.3 X 10 4 


(608 grams Be) in 




10 cm from the source, 


0.2 X 10 4 


graphite column, 




on the axis 




as above 




50 cm from the source, 
on the axis 


0.07 X 10 4 






100 cm from the source, 


0.008 X 10 4 






on the axis 





(b) The Center of a Nuclear Reactor ' 



Pile Type 


Location 


Power 
Level 
(kw) 


Flux 

(n/cm 2 • 
sec) 


Heterogeneous 
Graphite — U 


Oak Ridge, Tenn., U.S.A. 2 

Harwell, England (GLEEP) 3 
(BEPO) 4 


~10 3 
4X 10 3 


2 X 10 12 

10 8 
10 12 




Brookhaven, L.I., U.S.A. 5 


2.8 X 10 4 


4 X 10 12 


Homogeneous 
H 2 0— U 236 


Los Alamos, N. Mex., U.S.A. 
(HYPO) « 


6 


3 X 10 u 


Heterogeneous 
D 2 0— IT 


Argonne, 111., U.S.A. (CP3) ' 
Chalk River, Canada (NRX) ' 


300 
>10 4 


10 12 
5.8 X 10 13 




Chatillon, France (Zoe) 8 


—10 


3 X 10 10 




Oslo, Norway 9 


~100 


3 X 10 u 



Sec. 3B] Sources and Detectors 403 

TABLE 27 (Continued) 
Some Available Thermal Neutron Fluxes 

(c) Pile Thermal Columns 

Flux 
Pile Position (n/cm 2 • sec) 

Argonne (CP3) 10 inner face 3 X 10 s 

Chalk River (NRX) " 1 meter from inner face 10 9 

Chatillon (Zoe) 8 outer face 2 X 10 4 

(d) Holes in the Shield 



Pile 


Flux (n/cm 2 ■ sec) 


Oak Ridge 10 


10 7 


Chalk River (NRX) " 


4X 10 7 


Chatillon (Zoe) 8 


10 6 



1 D. J. Hughes, Nucleonics, 6 [2], 5(1950); H. S. Isbin, Nucleonics, 10 [3], 10 
(1952). 

2 A. H. Snell, Nucleonics, 8 [3], 3 (1951). 

3 F. C. W. Colmer and D. J. Littler, Nucleonics, 8 [1], 3 (1951). 
* Nucleonics, 8 [6], 36 (1951). 
6 Physics Today, Jan. 1951, p. 6. 

6 Nucleonics, 7 [6], 2 (1950). 

7 Nucleonics, 6 [3], 77 (1950); F. W. Gilbert, Nucleonics, 10 [1], 6 (1952). 
8 L. Kowarski, Helv. Phys. Acta, 23, Supp. 3, 70 (1950). 

9 Bulletin of the Atomic Scientists, 7, 380 (1951). 

10 Preliminary reports (unpublished) for NRC Subcommittee on Neutron 
Standards, by Bernstein and others. 

11 E. Almquist, Can. J. Research, A28, 433 (1950). 

12 D. G. Hurst, A. J. Pressesky, and P. R. Tunnicliffe, Rev. Sci. Instr., 21, 
705 (1950). 

the moderating medium. However, the beam intensities, which can be 
attained by this means (especially with easily available sources), are 
very low, since relatively large masses of moderating material are re- 
quired to thermalize the source neutrons, and the neutron densities, at 
the surface of the moderator, are consequently low; in addition, the 
emerging neutrons have a relatively broad angular spread, so that most 
of them are absorbed by the collimating slits. 

The common method of attaining strong thermal neutron beams is 
to use a "howitzer," which is a relatively large mass of moderator (usu- 
ally paraffin) with a hole reaching down to a region of high thermal neu- 
tron flux, close to the source. The hole is usually lined with cadmium 



404 The Neutron [Pt. VII 

or boron or both, thereby providing the necessary collimation for the 
beam. By proper design, it is thus possible to obtain relatively strong, 
well-collimated thermal neutron beams, even when only radioactive 
(a,n) sources are available (J 13). 

C. Neutron Detectors 

1. Induced Radioactivity. Neutrons are capable of inducing a variety 
of nuclear reactions, many of which result in radioactive product nuclei. 
Thus, neutrons can frequently be detected by the resulting radioactivity 
of the exposed substance. 

Radioactivity is detectable in many ways: Geiger counters for beta- 
rays of energy sufficient to penetrate the counter walls or, with lesser 
efficiency, for gamma-rays; electrometers or ionization chambers for 
alpha-, beta-, or gamma-radiation; scintillation counters, electron multi- 
pliers, etc. The feasibility of detecting the radioactivity resulting from 
neutron exposure depends on the lifetime of the induced radioactivity; 
it cannot be appreciably shorter than the time which must (for experi- 
mental reasons) elapse between exposure to neutrons and measurement 
of the resulting radioactivity; on the other hand, the lifetime must not 
be so long that the radioactive decay rate is negligible. These con- 
siderations normally have the effect of limiting the lifetimes of possible 
radioactivities to between ~10 — x sec and ~40 4 years. 

Consider a detector which absorbs neutrons at the rate of Rq per 
second, and whose consequent radioactivity decays with a mean life of 
r sec. The number of radioactive nuclei, N*, is governed by the differ- 
ential equation 

dN* N* 

— = R (100) 

dt r 

The activity of the detector (disintegrations per second), t sec after a 
(constant) exposure of duration t e , is 

R(t,t e ) = 7?o(l - e~ l ' /T )e- t/T (100a) 

The observed activity (usually counting rate), 

R'(t,Q = &R(t,t e ) (101) 

contains the factor S, the efficiency of the detecting system; S depends 
on the properties of the radiations involved, on the geometry of the 
absorber and detector, and on the efficiency of the radiation detector. 

The activity which would be observed immediately after an infinite 
irradiation, 

R'(t,t e ) 

R ' = #'(£ = 0, t e = oo) = __ _ = S R Q (ioia) 

(1 — e- ( «/'-)e _ '/ r 



Sec. 3C] Sources and Detectors 405 

is usually referred to as the "saturated activity." Its value is inde- 
pendent of the schedule of neutron exposure and radioactivity measure- 
ment. 

Ro' is usually measured by integrating the activity over a finite energy 
interval, say from t = t x to t - t 2 . The integrated activity is 

C(h,ta,Q = f t 'R'&Q dt = i\R'(ti,U) ~ R'(t 2 ,te)] (102) 

and the saturated activity is 

p , C(h,k,t e ) 

Ro — (102a) 

r(l - e- ( «A)(e- ( iA - e-'sA) 

The saturated activity is a measure of the neutron flux impinging on 
the absorber. If S is known, the radioactivity measurements yield R , 
which can be directly interpreted in terms of the neutron flux <j> = nv. 
We consider two types of measurement: 

(1) The absorber is embedded in an isotropic neutron flux whose 
spectrum is given by <j>(E), the density of neutrons per unit energy 
interval, at the energy E, multiplied by the neutron velocity correspond- 
ing to E. If, furthermore, the absorber is weak, 1 the rate of neutron 
absorption is 

R = Nvf<j>(E)<r a (E) dE (103) 

where N is the density of absorbing nuclei and V the volume of absorb- 
ing material. (NV is the total number of absorbing nuclei.) 

In general, both cf>(E) and a(E) are relatively complicated functions 
.of E, and a measurement of R serves only to determine the integral. 
However, there are some interesting special cases: Suppose that the 
neutron density has a Maxwell distribution 

*(») = vM(v) = Av 3 e - (v/v ' )2 (93a) 

and the cross section follows the 1/v law, 

<r a (v) = (94a) 

v 
Then 

R Q = NVc a 'v'A f v 2 e~ <»A') 2 dv 
NVAW)*tr a '7rX 



4 
1 By which we mean that its presence does not disturb the neutron density, 



(103a) 



406 The Neutron [Pt. VII 

The total flux is x 

$ = I M(v) dv = 
Jo 



A{v') 



iM 



whence 



2 



(93b) 



NV$a a 'ir 1A NV$c a ' 

R = = ( 10 3b) 

2 1.128 



Thus, for a Maxwell neutron distribution and al/» detector, the mean 
detector cross section is 



_ er a V IT 7 * , <f a 



fa = (95a) 

,v 2 1.128 

where a- a ' is the absorption cross section corresponding to the neutron 
energy E' = kT. 

Suppose, on the other hand, that the absorber has a single sharp 
resonance at the energy E r . (The effect of thermal neutron absorption 
can be eliminated by surrounding the detector with cadmium.) The 
cross section can be represented by a delta-function: 

c a {E) = P 8(E,E r ) (104) 

where 



p = j<r a (E)dE = -<r r 



(104a) 



for a resonance, of peak cross section <r and width r, which follows the 

Breit-Wigner formula. Provided that <t>(E) does not vary appreciably 

over the resonance, 

Ro = NV P <t>{E r ) (103c) 

When, as is frequently the case in the resonance region, 

4>{E) = - (96) 

Hi 

Eq. (103c) gives 

NVpB 
Ro = —=— (103c') 

E r 

If more than one resonance is involved, 

Ro = NVBYi ~ (103c") 

i Eri 

The expression 

E T = J v ~V~ (104b > 

is called the "resonance integral." 



Sec. 3C] Sources and Detectors 407 

Usually, the insertion of an absorber into the neutron flux causes a 
perturbation of the flux measured, so that the activity of the detector is 
not an exact measure of the true flux in the absence of the absorber. 
This perturbation arises from two causes: a reduction of the neutron 
density inside the detector which is due to the absorption of neutrons 
by the outside layers; and a depression of the flux directly outside the 
detector due to the finite neutron absorption. Such effects have been 
considered by Bothe (B53), who has derived a correction factor /for the 
reduction of the specific activity of a sphere of radius R, 



1 a 

- = 1 +- 
/ 2 



3RL 

~2\^R+~L)~ . 



for R » \ ir (105a) 



1 0.34aK 

- = 1 + — for R « X tr (105b) 

J A«r 

In the above expressions, L is the diffusion length (see Section 4) of the 
neutrons in the medium outside the detector; \ tr is their transport mean 
free path; a is the average probability that a neutron will be absorbed 
in a single traversal of the detector. For an isotropic flux 

■ « = 1 _ e -» T {l - pT) + n 2 T 2 Ei{-».T) (105') 

where ft = N<r a is the absorption coefficient, in cm -1 , of the detector 
and T is the average absorber thickness. (T = R for a spherical ab- 
sorber.) Ei{ — x) is the exponential integral. 

Tittle (T12) has shown experimentally that the above expressions 
also apply without modification to an absorber in the form of a disk of 
radius R and thickness T. 

(2) The second situation of interest involves neutron beams. Con- 
sider a neutron detector of area (perpendicular to the beam direction) 
A and thickness T. The absorption rate is 



Ro = aU(E)(1 - e-" r ) 



dE 



(103') 



An absorber for which fiT = NTc a (E) » 1 is said to be "black" for 
neutrons of energy E. Since 

ft (black) = A J 0(2?) dE (103 'a) 

such an absorber gives a direct measure of the total neutron flux. At 



408 The Neutron [Pt. VII 

the other extreme, a "transparent" detector is one for which fiT <JC 1, in 
which case 

R (transparent) = ATN f<t>(E)* a (E) dE (103'b) 

Owing to the large variation with energy of slow-neutron cross sec- 
tions, most absorbers are transparent at some energies, black at some, 
and translucent at others. 

2. The Szilard-Chalmers Reaction. In many nuclear reactions, e.g., 
(n,p), (n,a), (ft,/), the product nuclei differ in their chemical properties 
from the target nuclei, and can therefore be separated from the target 
with relative ease. For such detectors the radioactivity from large 
masses of absorber can be reduced to moderately small samples for 
counting, and the absolute detection efficiency can be made relatively 
high. However, many of the important neutron reactions — (n,y), {n,n'), 
(n,2n) — lead to isotopes of the target element. For such detectors the 
specific activity of the absorber may be too small to permit accurate 
measurement of relatively weak neutron fluxes. 

To overcome these difficulties Szilard and Chalmers (S62) in 1934 
devised a technique for separating radioactive nuclei, induced by the 
(n,y) reaction, from their isotopic environment. This technique takes 
advantage of the fact that the emission of gamma-radiation after slow- 
neutron capture causes a recoil of the product nucleus; the recoil energy 
is usually sufficient to disrupt the molecular bond and, thereby, to 
change the chemical state of the product nucleus as compared to a nor- 
mal nucleus in the medium. The radioactive nuclei can then, in favor- 
able circumstances, be separated chemically from the normal nuclei, 
with efficiencies as high as 100 percent. 

For the emission of a single gamma-ray of energy E y (in Mev) from a 
nucleus of mass number A, the nuclear recoil energy E (in ev) is 

536E y 2 , N 

E = (106) 



For typical values, E 7 = 7.5 Mev and A = 100, the recoil energy is E 
= 300 ev, which is considerably in excess of that usually necessary to 
disrupt chemical bonds, say ~5 ev. Even if two or more gamma-rays 
are emitted after slow-neutron capture, all but a few percent of the re- 
coiling nuclei will usually have E > 5 ev. 

In their original experiment Szilard and Chalmers irradiated ethyl 
iodide with neutrons from a Ra-a-Be source (most of the absorptions 



Sec. 3C] Sources and Detectors 409 

are due to slow neutrons) and extracted the radioactive I 128 (25-min 
half-life) by shaking with an aqueous reducing solution containing a 
trace of free iodine as a carrier. Similar techniques have been applied 
to other halogens, and methods have been worked out for separating 
the product nuclei, due to the (n,y) reaction, of many other elements. 
Specific Szilard-Chalmers processes for different substances are de- 
scribed and discussed in excellent summaries (with references) of the 
technique by McKay (M14), by Broda (B68), and by Barnes, Burgus, 
and Miskel (W2). 

The Szilard-Chalmers reaction has frequently been used to obtain 
radioactive samples of high specific activity, especially when only rela- 
tively weak neutron sources were available. Thus, von Halban, Kowar- 
ski, and Magat (H6) employed a solution of bromine to measure the 
intensity of neutrons in the cosmic radiation. The strength of relatively 
weak fast-neutron sources is frequently measured by stopping the 
neutrons in a water solution of potassium permanganate, and removing 
the radioactive Mn 56 (2.6-h half-life) as Mn0 2 by filtration through 
fine paper or sintered glass (E6) ; the efficiency of this Szilard-Chalmers 
reaction is greatest for a neutral or slightly acid solution. Broda and 
Rieder (B67) have shown that the same reaction can be used to detect 
fast neutrons. 

3. Ionization Chambers and Proportional Counters. Many neutron 
reactions involve the prompt emission of a charged particle. Such reac- 
tions can be detected, through the ionization caused by the emitted 
particle, in an ionization chamber or a proportional counter. In addi- 
tion to whatever other distinctive properties such detectors may possess, 
they have the advantage over induced radioactivity that they can be 
used to observe neutrons in situ during the irradiation, and will reflect 
short-time fluctuations in the neutron density; a radioactive detector, 
on the other hand, is inherently a time-integrating device. 

Owing to the relatively short range of the heavy charged particles 
resulting from neutron reactions, the target nuclei must, in general, 
be inside the chamber. They can be introduced into the chamber 
either as a constituent of the gas with which the chamber is filled (and 
in which the detected ionization takes place) or as a coating on the 
inside walls. 

In ionization chambers the ionization of the gas is measured (without 
amplification in the chamber) either as a current or, in the case of pulsed 
ionization chambers, as individual pulses associated with single nuclear 
disintegrations. Proportional counters differ only in that the ionization 
is amplified in the gas of the chamber. In both cases, the observed 
charge or pulse height is proportional to the energy dissipated in the 



410 The Neutron [Pt. VII 

chamber (although the constant of proportionality may vary somewhat 
with the position in the chamber at which the event occurs). 

The construction and use of ionization chambers and proportional 
counters has been discussed in Part I of Volume I. The following is 
intended as a brief summary of the main considerations involved in the 
use of such devices to measure neutron fluxes: When a neutron-detecting 
ionization chamber or proportional counter is placed in a neutron flux 
<j>(E), the observed intensity, / (ionization rate or counting rate), is 
determined by the same factors as the saturated activity R ' of a radio- 
active detector (Section 3C1) with the important difference that the de- 
tection efficiency is, in general, a function of the neutron energy. Thus, 
we have (neglecting the variation of efficiency with position in the cham- 
ber) : for a chamber in an isotropic neutron flux, 

I = NV f&(E)^(E)a(E)f(E) dE (103") 

where the symbols are as previously defined and f(E) is given by Eqs. 
(105a), (105b), (105'); for a chamber in a neutron beam, 

I = A £&(E)<}>(E)(1 - e-* r ) dE (103'") 

The efficiency function, &(E), depends on the particular reaction used, 
on the chamber geometry, on the method of introduction of the reacting 
nuclei, and on the particular detection method used. The following dis- 
cussion applies only to neutron reactions in which all the energy is car- 
ried away by charged particles. Reactions involving the emission of 
neutrons as well as charged particles (of which the most important is 
neutron scattering, especially by protons) will be taken up in a separate 
section. We consider a number of the most widely used detection 
schemes : 

(la) / = ionization current; the reacting nuclei are in the chamber 

gas: 

8(E) = XX (E + Q) X G{E) (107) 

where E is the neutron energy and Q is the reaction Q value. (Note : For 
Q < 0, S = for E < -[(A + 1)/A]Q.) G(E) is a geometric factor 
which depends on the chamber shape, on the gas pressure, and on the 
range-energy relationship for the emitted charged particles in the cham- 
ber gas. It represents the average (over the chamber) fraction of the re- 
action energy which is dissipated in the chamber gas; G(E) — > 1 as the 
range of the emitted particles becomes small compared to the chamber 
dimensions. The calibration constant K can be determined by measur- 



Sec. 3C] Sources and Detectors 411 

ing the ionization produced by a known flux of known energy; alterna- 
tively, K can be determined by observing the ionization resulting from 
the insertion into the chamber of a calibrated source of polonium alpha- 
particles. 

(lb) / = ionization current; the reacting nuclei are introduced as a 
coating on the chamber walls. If the coating is thin compared to the 
range of the emitted charged particles, the situation is similar to that 
for a gas-filled chamber, except that only ~ \ the reaction energy is dis- 
sipated in the chamber gas. As the coating thickness is increased, the 
emerging charged particles will lose energy in traversing the coating. 
For a thick coating (T > range of the emitted charged particles) only 
those reactions taking place within a layer of thickness equal to the 
range of the emitted charged particles can be effective in producing 
ionization in the chamber gas. The efficiency function of an ionization 
chamber with a given coating can, in general, be computed. In addition 
to its dependence on the specific properties of the reaction, the effi- 
ciency may also depend on the angular distribution of the neutron flux 
being detected. 

The form of &(E) is quite different for the counting of individual 
pulses. In general, the associated electronic equipment is designed so 
that only pulses of magnitude greater than a predetermined value are 
counted. (This type of discrimination makes it possible to use pulse 
counters even when the neutron flux is accompanied by a considerable 
background of gamma-radiation.) A pulse detector in which all pulses 
greater than a given size are counted is called an "integral" detector. 
It is also possible to arrange the electronics so that only those pulses 
are counted whose sizes lie between two definite (usually variable) 
limits; such arrangements are called "differential" detectors. 

(2) I = counting rate; integral detection. If the reacting nuclei are 
in the chamber gas, &(E) is, as a first approximation, given by a step 
function, having the value zero for neutron energies less than E t , and 
one for E > E t ; E t is the neutron energy for which the ionization of the 
reaction products is just sufficient to produce a pulse of the required 
size. However, in a chamber of finite size the step function must be 
modified by a geometric factor G(E), to take into account the possibility 
that some products of an otherwise detectable reaction may dissipate 
part of their energy in the chamber walls and give rise to pulses smaller 
than the detectable minimum. 

When the target nuclei are coated on the chamber walls the same 
considerations apply as in scheme (lb). 

(3) / = counting rate; differential detection, with the reactions occur- 
ring in the gas of the chamber. Ideally (for a chamber of infinite dimen- 



412 The Neutron [Pt. VII 

sions) &(E) is given by the difference between two step functions with 
thresholds E t \ and E t 2- In practice, the effect of the finite size of the 
chamber is to make S smaller than 1 between these limits and to add to 
&(E) a tail extending beyond E t2 . 

When the reacting nuclei are introduced as a wall coating, differential 
detection is useful only if the coating thickness is small compared to the 
range of the reaction products. 

(a) Detection by the B 10 (n,a) Reaction. Because of its large cross sec- 
tion, the B 10 (w,a) reaction is extensively used in slow-neutron detectors. 
The reaction is highly exoergic, Q = 2.78 Mev. However, most slow- 
neutron captures lead to the 0.48-Mev excited state of Li 7 (which decays 
to the ground state by gamma-ray emission); the ionization energy re- 
leased in such captures is 2.30 Mev. Bichsel, Halg, Huber, and Stebler 
(B29) and Petree, Johnson, and Miller (P14) have measured the ratio of 
captures leading to the ground and to the 0.48-Mev excited states for 
neutron energies from thermal to 3.9 Mev. The ratio follows a smooth 
curve, from ~0.07 for thermal neutrons to a maximum of somewhat 
greater than 2 at 1.9 Mev, and then falls to ~1 above 2.5 Mev. 

The (n,a) cross section of normal boron element follows the 1/v law 
up to energies of at least 1 kev, with <r'(y' = 2.2 X 10 5 cm/sec) = 710 
barns. Its behavior in the intermediate range is not so well established, 
but its general features have been determined; there is a resonance at 
1.9 Mev (G20, A2, P14). 

The isotope B 10 has a natural abundance of 18.83 percent. Its iso- 
topic (n,a) cross section is greater than that for natural boron by the 
factor 5.31 ; i.e., </(B 10 ) = 3770 barns. 1 The availability of B 10 enriched 
boron thus makes possible the construction of neutron detectors with 
high efficiency over a wide range of neutron energies. 

Boron-containing ionization chambers and proportional counters have 
been operated under a variety of conditions, as current-measuring de- 
vices and disintegration counters, with the boron introduced as a gas 
or as a wall coating. Because of the large Q value, 8(E) is essentially 
energy independent in the slow- and intermediate-neutron energy 
ranges. Furthermore, the large Q value makes it easy to discriminate 
between pulses resulting from neutron capture and background pulses 
arising from relatively strong gamma-ray intensities; hence, boron- 
containing chambers are normally operated as integral pulse counters. 

BF 3 is usually used as the chamber-filling gas. It has relatively good 
ionization chamber and proportional counter characteristics, provided 
that the gas is free of impurities. Techniques for the construction, fill— 

1 See the footnote to Table 11, p. 339. 



Sec. 3C] Sources and Detectors 413 

ing, and operation of BF 3 counters have been discussed by a number of 
authors (S15, B31, F39, B56). 

For a chamber containing BF 3 at one atmosphere pressure, the slow- 
neutron absorption coefficient is 

M (normal BF 3 ) = 0.0191 (0.025/E)' A cm -1 

(i (B 10 F 3 ) = 0.101 (0.025/^)* cm -1 

(E is in ev.) Thus, a typical counter (of average thickness ~25 cm) 
would vary in effectiveness from black to transparent over the ther- 
mal neutron range. However, most BF 3 counters — even B 10 F 3 — are 
transparent for epithermal neutrons. For a slow-neutron-induced dis- 
integration in such a chamber, the range of (93% of) the alpha-particles 
is -~0.8 cm. (The Li 7 fragment, which carries off f{ i of the energy, has 
a much smaller range.) Thus, for chambers of reasonable dimensions, 
the wall effect correction G(E) is relatively small, and &(E) « 1. 

Chambers in which the boron is introduced as a wall coating are less 
extensively used for slow-neutron detection. The thickness of the coat- 
ing must be kept very small, both because of the small range of the reac- 
tion products and because of the strong neutron absorption of boron (if 
the neutrons must penetrate the coating from the outside). Hence, such 
chambers are usually quite transparent to neutrons. It is possible to 
decrease the transparency, without decreasing the efficiency, by design- 
ing the chamber to contain a multiplicity of thin boron layers (L24). 

(b) The Long Counter. The sensitivity of BF 3 counters decreases 
rapidly with neutron energy, becoming prohibitively small, for most 
counters, in the intermediate-energy range. In order to increase the 
sensitivity to intermediate and fast neutrons, experimenters have fre- 
quently immersed the counter in a moderating medium, usually water 
or paraffin. However, because of the strong dependence of the sensi- 
tivity of such arrangements on the source energy and on the geometry, 
the results of this method of neutron flux measurement are usually very 
difficult to interpret, especially when the source has a heterogeneous 
energy distribution. 

Hanson and McKibben (H32) have devised an arrangement which is 
uniformly sensitive to neutrons from ~10 kev to ~3 Mev. This ar- 
rangement, the "long counter," consists of a cylindrical BF 3 counter 
(diameter ~J in., length ~10 in.) placed along the axis of a paraffin 
cylinder ~8 in. in diameter. Two of their successful long counter de- 
signs are shown in Fig. 47. 

The long counter is used in observations on neutron beams which 
enter the circular face (right-hand side in Fig. 47a) of the arrangement 




preamplifier 



Aluminum tube 



Long Counter, 8"0.D. 



Case of 0.050" sheet iron. 



Removable aluminum 
cylinder containing 
BF, tube 



High voltage 




-3'^) J* 



Shielded Long Counter 
(a) 




2 3 4 

Neutron energy (Mev) 

(b) 
Fig. 47. (a) Two long counters, designed by Hanson and McKibben (H32), with 
uniform sensitivity (to within ~10 percent) from 10 kev to 3 Mev. The second, or 
shielded, counter is designed to operate in a region of relatively high scattered neu- 
tron background, (b) Sensitivity vs. neutron energy of the shielded long counter 
with and without holes in the front face of the paraffin: 
414 



Sec. 3C] Sources and Detectors 415 

and diffuse parallel to the axis. The second arrangement shown is de- 
signed for situations in which there is an appreciable background of 
scattered neutrons impinging on the paraffin from all directions. The 
uniform energy sensitivity is based on the approximate cancellation of 
two effects: (1) low-energy neutrons do not penetrate far into the paraf- 
fin, and have an appreciable probability of being turned around and 
escaping out of the front face; (2) high-energy neutrons penetrate much 
farther before being slowed down, but have an appreciable probability 
of escaping through the sides. The success of the arrangements of Han- 
son and McKibben is based on a proper choice of the diameter of the 
paraffin cylinder. In the second design, holes can be bored into the front 
face to decrease the loss of low-energy neutrons (Fig. 47b). 

For the shielded counter shown in Fig. 47a, the sensitivity does not 
vary by more than ~10 percent between 10 kev and 3 Mev, as shown in 
Fig. 47b. At lower energies the sensitivity decreases to -~70 percent for 
thermal neutrons; at higher energies the sensitivity again falls off (it is 
~85-95 percent for Ra-a-Be neutrons). The absolute sensitivity of 
the counters is such that they give ~1 count per 10 s neutrons emitted 
isotropically from a source 1 meter from the face. 

(c) Fission Chambers. Fission of the heaviest nuclei by neutrons 
provides an ideal source of ionizing particles for pulse detectors. Since 
the fission fragment energies are -~100 Mev and their ranges are very 
small, the pulse due to a fission fragment is easy to distinguish from the 
background due to other ionizing radiation (e.g., pulses due to alpha- 
particles, spontaneously emitted from most fissionable nuclei). Thus, 
fission chambers are almost always operated as integral pulse-counting 
ionization chambers. 

The fissionable nuclei may be introduced into the chamber as a gas 
(e.g., UF 6 ) or as a wall coating. Owing to the general unavailability of 
gases containing the heaviest elements or the inconvenience in handling 
these gases, or both, the latter method is usually used; uranium, for ex- 
ample, is easily obtainable in a solid oxide or nitrate form. In either 
event, the efficiency function, &(E), is essentially independent of neutron 
energy, since the Q value of the (n,f) reaction is so great. (We neglect 
the possibility that the coating may be so thick as to prevent the neu- 
trons from penetrating into the chamber.) 

Fission chambers containing thermally fissionable nuclei (U 233 , TJ 235 , 
Pu 239 ) are efficient thermal neutron detectors. Chambers containing 
natural uranium, or uranium enriched in the U 238 isotope, can be used 
as fast-neutron detectors, with an effective threshold of ~1.5 Mev; 
thorium, protactinium, and neptunium can also be used in the same 
fashion (see Section 2C4). 



416 The Neutron [Pt. VII 

Bismuth undergoes fission by ultrafast neutrons; it has a threshold of 
~50 Mev and a cross section, for ~84 Mev neutrons, of ~0.05 barn 
(K7, W26). Ionization chambers coated with bismuth have been used 
as specific ultrafast-neutron detectors (W26). 

(d) Proton Recoil Detectors. Perhaps the most common method of 

detecting and measuring fast-neutron fluxes involves the observation of 

proton recoils from neutron-proton scattering. Use of this reaction has 

the advantage that the cross section is comparatively large (14 barns 

at E = 0.1 Mev; 4 barns at E = 1 Mev; 1 barn at E = 10 Mev; see 

Fig. 2). The main disadvantage of the proton recoil method is that for 

a given incident neutron energy, E n , the proton recoil energies, E, are 

uniformly distributed between and E n ; the recoil energy as a function 

of the angle 6 in the laboratory coordinate system (0 is confined to the 

interval 0-90°) is 

E = E n cos 2 6 (108) 

Let us assume a monoenergetic neutron flux <j>(E n ) incident on a trans- 
parent chamber containing NV atoms of hydrogen. Neglecting, for the 
moment, the effects of the finite dimensions of the chamber, the number 
of pulses corresponding to the proton energy E is 

NV<j>(E n )<r(E n ) cLE 

P(E n ,E) dE = VK I ~ (109) 

E n 

for E <E n ; P(E n ,E) = for E > E n . 

The chamber may be operated as an integral counter, in which all 
pulses are counted whose energy exceeds the bias energy E t . In this 
case, the counting rate is 



E " IV(tf n ) 



I(E n ,E t ) = f P(E n ,E) dE = NV4>(E n ) 

JBt 



(E n - E t ) (110) 



Proton pulses of energy less than ~0.1 Mev are usually too small to 
be distinguished from the background. In the region 0.1-10 Mev, the 
neutron-proton scattering cross section follows, to a good approximation, 
a l/v law; i.e., <r(E n ) ^ (E t /E n ) y2 <r(E t ). In this energy range 



I(E n ,E t )^NV<j>(E n )a{E t ) (—J 
= NV<j>(E n )<r(E 



E n 
l~E t ~ l 



«Kt) 



(110a) 



The sensitivity of such an integral pulse counter is plotted as the solid 
curve in Fig. 48. S(E n /E t ) has the property, in this energy range, of 



Sec. 3C] 



Sources and Detectors 



417 



being relatively independent of the neutron energy; its value is within 
30 percent of the maximum for 1.52^ < E n < U.5E t . 

For neutrons of E n > 10 Mev, the neutron-proton cross section can 
be approximated by <r(E n ) = {E t /E n )c{E t ), whence 



«®-®'[®-' 



(110a') 



This sensitivity function is plotted as the broken curve in Fig. 48. In 



1.0 



f 

I 0.! 



I 0.6 

8 



0.4 





7 

/ 
1 i 


f\ 






































II 
-if 




\ 
\ 




































1/ 

41 H 




S 


\ 




































1 






> 


































If 
(J 








v 


X 






























1 














•». 


























1 
li 
















^». 


-- 


->■ 


















\\ 


























-— 


-- 

























































































— ~-r=r 0.2 



! 10 



16 



18 



Tig. 48. The sensitivity function S(E n /E t ) of an integral pulse-counting hydrogen- 
filled chamber (neglecting wall effects) vs. x = E n /E t . The solid curve is for <r(E n ) 
= (Ei/E n ) ' A cr(Et), a good approximation to the neutron-proton scattering cross 
section for the range 0.1 Mev < E n < 10 Mev. The broken curve is for a{E n ) 
= (E t /En)<r(Et), which applies for E n > 10 Mev. The curves are normalized to a 
value of 1 at the maximum. 

this energy range the integral counting rate is much more strongly de- 
pendent on E„, However, for neutron energies above ~20 Mev, the 
angular distribution in neutron-proton scattering is no longer spherically 
symmetrical in the center-of-mass coordinate system, and the formulas 
developed above must be correspondingly modified. 

Hydrogen-filled chambers are frequently operated as ionization cham- 
bers. For such operation the chamber sensitivity to neutrons of energy 
E n is 



I'{E n ) = I 1 

J n 



P(E n ,E)E dE = 



NV4>{E n )*(E n )E n 



(111) 



for neutrons of energy up to ~20 Mev, and neglecting wall effects. How- 
ever, ionization chambers are particularly sensitive to backgrounds of 
all kinds. Since most neutron fluxes are accompanied by an appreciable 



418 The Neutron [Pt. VII 

gamma-ray intensity, the background presents a serious problem. One 
device for eliminating such background effects is to employ two chambers 
of identical geometry, one filled with the hydrogen-containing gas and 
the second filled with argon. The external electronic amplifications are 
adjusted in a pure gamma-ray beam so that both chambers give the 
same current. The difference in the currents, when the chambers are 
placed in a neutron flux, can be attributed to the proton recoils from 
neutron scattering. 

The effect of the finite dimensions of the chamber is to decrease the 
sensitivity by a factor which depends on the neutron energy and which 
can be computed for most chambers. The geometric factor, G(E n ), is a 
monotonically decreasing function of E n . Unfortunately, the recoil 
proton ranges are in general comparable to the chamber dimensions, so 
that G(E n ) usually represents a sizable correction. For a given cham- 
ber geometry the correction becomes less important as the gas pressure 
is increased. Thus, the development of techniques for the operation of 
ionization chambers and counters at high pressure (W35) is of consider- 
able importance for the future of fast-neutron spectroscopy. Alterna- 
tively, since organic phosphors (solid and liquid) have a high hydrogen 
content, it is possible to employ such materials for fast-neutron counting 
by observing the scintillations resulting from the recoil protons (J3, 
06) ; x the recoil proton range is usually small compared to the dimen- 
sions of the phosphor. Unfortunately, the pulses from such counters do 
not appear to be strictly proportional to the recoil proton energy. 

The above discussion applies to monoenergetic neutron sources. If 
the neutron source is heteroenergetic, the pulse height distribution is a 
superposition of pulses due to all the neutrons : 

r°° r x <t>(E n )a(E n ) dE n 

P(E) dE = I P{E n ,E) dE n dE = NV dE I — — — — (109') 

Je J e E n 

The function <j>(E n )a(E n )/E n can be obtained by differentiation of the 
curve of P(E) vs. E. However, in order to determine <t>(E n ) vs. E n to a 
reasonable accuracy, the curve of P(E) vs. E must be determined to a 
much greater — frequently unattainable — accuracy. The use of integral 
counting to obtain 4>{E n ) vs.'E n is even less satisfactory in that it re- 
quires exceptionally high precision. 

When the incident neutrons are in a beam (unique direction of motion) 
it is possible to choose, for counting, only those proton recoils whose 
directions are the same as that of the incident neutrons. In such a de- 

1 An ingenious application of this principle has been devised by W. F. Hornyak, 
Rev. Sci. Instr., 23, 264 (1952). 



Sec. 3C] Sources and Detectors 419 

vice it is necessary to collimate the protons. However, the collimation 
need not be too sharp; a proton recoil at, say, 20° from the incident 
neutron direction has an energy E = 0.88^„ (Eq. 108). Assuming a 
collimating system which accepts protons within the relatively small 
angle 6 , and neglecting the small variation of pulse heights within the 
cone of acceptance, 

P(E) = NV<KE)*(E)f(fi ) (109") 

where 

AE 6 2 

f(!h) = — = 1 - cos 2 O = — (108a) 



(d may be a function of the proton recoil energy E.) The smaller the 
angle do, the smaller is the value of f(ff), but the sharper the energy reso- 
lution of the device. Thus, in common with almost all problems in 
spectroscopy — neutrons or otherwise — the practically attainable reso- 
lution is primarily determined by the strength of the source. 

Many arrangements can be conceived for collimating the observed 
protons. One possibility is to detect the protons in a long cylindrical 
chamber, the incident neutrons traveling along the chamber axis, and 
to divide the chamber into two or more independent regions, requiring 
coincidences between adjoining regions. In some cases, the last section 
is operated in anti-coincidence with the preceding sections, thereby de- 
fining the range of the detected protons. Another possibility is to 
separate the sections by barriers, with holes in them for the proton 
collimation. The hydrogen may be introduced into the chamber as a 
gas or, more usually, as a thin radiator at the incident face of the cylinder. 

A thin radiator is, by definition, one in which a proton, originating in 
the back of the radiator, loses only a small fraction of its energy before 
emerging into the counter. The figures in Table 28 give the approxi- 

TABLE 28 
Some Typical Proton Recoil Ranges 







Range 


Range 


E p 


Range 


(mg/cm 2 


(mm 


Mev) 


(cm std. air) 


paraffin) 


paraffin) 


1 


2.3 


3.2 


0.036 


2 


7.1 


9.7 


0.11 


5 


34 


47 


0.52 


10 


115 


160 


1.8 


15 


238 


330 


3.7 



mate ranges of fast protons in standard air and in paraffin, a typical 
radiator material. It may be seen that, for neutrons of energy ~1 Mev 



420 The Neutron [Pt. VII 

or less, "thin" radiators must be very thin indeed. As the radiator thick- 
ness becomes comparable to the proton range, the energy resolution of 
the detector rapidly deteriorates. For thick radiators (thickness > 
maximum proton recoil range) the situation again arises wherein a 
monoenergetic neutron produces pulses of all energies up to its own. 
However, the pulse height distribution is much more complicated than 
that from neutrons scattered in the chamber gas. For neutrons of 
energy E n incident normally on a thick radiator, with the chamber 
counting all pulses above the bias energy E t , the sensitivity function is 
approximately (B9) 

S(E n ,E t ) g* ke(E n )E n - y \E n y> - E t %A f (110a") 

in which the range-energy relationship has been approximated by 
B(E P ) oc E P 3A . This function has zero slope at E n = E t , and is a mono- 
tonically increasing function of E n ; hence, such a chamber is most sensi- 
tive to the highest-energy neutrons in the beam. 

(e) Other Possibilities. In principle, any neutron reaction can be used 
as the basis for a neutron detector. Thus, the radiative capture process 
can be observed by detection of the prompt capture gamma-rays; in- 
elastic scattering can be detected by observation of the accompanying 
gamma-radiation. With the recent development of scintillation counter 
techniques, the efficiency of gamma-ray detection is comparable with 
that for the detection of charged particles. 

One application of these techniques to neutron counting is of special 
interest in that it can be used to provide an efficient detector of inter- 
mediate-energy neutrons (D16). This technique takes advantage of the 
fact that, for slow and intermediate neutrons, the B 10 (w,a) reaction goes 
mainly to the 0.48-Mev excited state of Li 7 , from whose decay the 
gamma-ray can be detected by a scintillation counter. Since, in such a 
detector, solid boron can be used, the size being limited only by the 
penetrability of the 0.48-Mev gamma-ray, high efficiencies are easily 
obtainable. 

As previously pointed out, charged particle reactions are easily ob- 
servable if they take place in a scintillating material. Thus, Hofstadter 
et al. have detected thermal neutrons, with high efficiency, by the scin- 
tillations produced, in a Lil(Tl) crystal, as a result of the Li 6 (n,a) 
reaction (H64). Such reactions can also be observed by allowing the 
charged particle to impinge, from the outside, on a scintillator. The 
zinc sulfide screen, so important in the pioneer work on natural radio- 
activity, has been used in this connection (M38). 

Other charged particle reactions, both exoergic and endoergic, can be 
used for ionization chamber and proportional counter materials. Among 



Sec. 3C] Sources and Detectors 421 

these, the N 14 (n,p) reaction, Q = 0.626 Mev, has possibilities as an 
intermediate-neutron detector, since its relatively low Q value permits 
identification of the energy of the neutron responsible for the reaction 
through a pulse height measurement (F7). Unfortunately, the presence 
of resonances in this reaction, for neutron energies above ~500 kev, 
limits its usefulness to intermediate-energy-neutron spectroscopy. 
Other reactions with similar characteristics are He 3 (n,p) and Cl 3S (n,p). 

Finally, recoils from elastic scattering by light nuclei other than H 1 
can be used for neutron detection. In this regard, He 4 recoils have the 
advantage of shorter range (thereby decreasing the wall effect correc- 
tion) and less energy spread for a given incident neutron energy. 

4. Photographic Emulsion and Cloud Chamber Detectors. Photo- 
graphic emulsions and cloud chambers have been among the most use- 
ful tools for the observation of nuclear reactions. Both of these devices 
have the property that the path traversed by a charged particle appears 
as a visible track. From the length of the track (range) and the density 
of ionization (grain or droplet density), it is possible to determine the 
energy and velocity, respectively, of the ionizing particle. A curve of 
ionization density vs. residual range for a track uniquely determines the 
energy, mass, and magnitude of charge of the particle. Measurement 
of the multiple Coulomb scattering in the photographic emulsion or in 
the gas or plates in a cloud chamber provides a measure of the particle 
energy. The curvature of the track in a known magnetic field yields the 
particle's momentum, as well as the sign of its charge. Since track 
curvature results both from small-angle Coulomb scattering and from 
applied magnetic fields, the use of magnetic fields is usually confined to 
cloud chambers, or to regions between two photographic emulsions, 
since the Coulomb scattering is smaller and the range is greater in a gas 
than in the emulsion; however, for sufficiently strong fields and particles 
of high energy, curvature due to magnetic fields has been observed on 
tracks in emulsions (D14). 

In essence, photographic emulsions and cloud chambers yield the same 
sort of information; indeed, the photographic emulsion can be said to be 
identical with a cloud chamber operated at very high pressure. How- 
ever, owing to the differences in their mode of operation, emulsions and 
cloud chambers should be regarded as complementary instruments, the 
choice between the two being primarily determined by the nature of the 
problem under investigation. Cloud chambers have the advantage that 
it is easier to choose and change the filling gas. On the other hand, 
emulsions are continuously sensitive, and are therefore much more use- 
ful in experiments involving weak sources. Furthermore, the sensitivity 
of emulsions is variable over a very wide range — emulsions are available 



422 The Neutron [Pt. VII 

which are insensitive to tracks of ionization density less than that of a 
fission fragment; at the other extreme, it is possible to obtain emulsions 
which show tracks of electrons at the minimum of the ionization rate 
curve. Another advantage of emulsions is their small size and light 
weight. 

The main disadvantage in the use of photographic emulsions arises 
from the short range, in them, of charged particles. (A 1-Mev proton 
has a range of ~15 /*, a 1-Mev alpha-particle of ~3.5 /j..) As a result, 
it is necessary to detect and measure tracks with a microscope, and the 
accumulation of large amounts of data becomes a relatively tedious 
affair compared to experiments involving electronic detection. How- 
ever, the wide flexibility and the possibility of unambiguous identifica- 
tion of reactions frequently more than compensate for this disadvantage. 

The earliest experiments with photographic emulsion detection of 
charged particles employed x-ray and "halftone" plates. Thus, Taylor 
and Goldhaber (T4), as early as 1935, used boron-impregnated Ilford 
R emulsions, irradiated by slow neutrons, to prove that the B(n,a) reac- 
tion involves the emission of a single alpha-particle. However, the 
emulsions available at that time could only detect low-energy charged 
particles; their lack of sensitivity and uniformity greatly discouraged 
their use in nuclear physics. Since World War II, tremendous progress 
has been made in the development of more sensitive emulsions and of 
techniques for their use, in large measure through the instigation and 
inspiration of Occhialini, Powell, and their collaborators. Emulsions 
called "nuclear emulsions," with various degrees of sensitivity, and with 
various thicknesses, from 25 to 600 fi, are now available. (Thicker emul- 
sions can be obtained by special arrangement with the manufacturers.) 
The use of nuclear emulsions has been extensively described by Powell 
and Occhialini (P25), by Yagoda (Y2), and in a recent summary by 
Rotblat (R22). Rotblat also gives the most recent data on the composi- 
tion and properties of the various available emulsions (his Tables I, II, 
and III) and on processing techniques. 

There is an extensive literature on cloud chambers. A recent mono- 
graph by Wilson (W34) summarizes the techniques of cloud chamber 
operation. Further details on these subjects are given in Volume I, 
Part I. 

(a) Detection of Slow Neutrons. Nuclear emulsions contain an appre- 
ciable amount of nitrogen (0.073, 0.080, 0.11 g/cm 3 , respectively, in Il- 
ford, Kodak, and Eastman Kodak emulsions). The N 14 (n,p) reaction 
has a Q value of 0.626 Mev, with a cross section a' = 1.76 barns for 
thermal neutrons. The protons resulting from thermal neutron capture 



Sec. 3C] Sources and Detectors 423 

(range ~7 n) can be observed and counted in the emulsion, thereby 
providing a measure of the thermal neutron flux (C33). 

The sensitivity of nuclear emulsions to slow neutrons can be greatly 
enhanced by adding small amounts (plates are available with ~1% by 
weight) of lithium or boron to the emulsions. The (n,a) reactions in 
such loaded emulsions can be used for measuring even small slow-neutron 
fluxes with relatively high accuracy (S25, T10, B35, R22). 

It is also possible to impregnate emulsions with uranium acetate or 
citrate, and to observe tracks due to fission by slow neutrons (M43). 
Owing to the high background of alpha-particle tracks from the uranium, 
such plates must ordinarily be exposed and processed within a short 
time after preparation. However, emulsions are available whose sensi- 
tivity to alpha-particles has been greatly decreased, so that only the 
fission fragment tracks are observable (Ilford Dl, Eastman Kodak 
NTC). 

It is also possible to measure slow-neutron fluxes by observing the 
blackening of x-ray and electron sensitive emulsions resulting from slow- 
neutron-induced radioactivity (B35). 

The reactions mentioned above can also be observed in cloud cham- 
bers, although it is considerably more difficult to obtain a suitable gas 
containing lithium, boron, or uranium; instead, these elements are 
usually introduced into the chamber as thin coatings on plates. 

(b) The Observation of Proton Recoils. Nuclear emulsions contain an 
appreciable hydrogen content (~0.04-0.05 g/cm 3 in dry emulsions; 
considerably more in a moist atmosphere). Thus, irradiation with fast 
neutrons will give rise to proton recoil tracks. 

Because of the variation of the proton recoil energy with angle, accord- 
ing to Eq. (108), measurement of the energy of a proton recoil will yield 
the neutron energy only if the relative directions of neutron and proton 
are known. In other words, to use proton recoils for neutron energy 
and flux measurements, it is necessary for the incident neutrons to have 
a well-defined direction. (The previous discussion of proton recoil 
ionization chambers and proportional counters is fully applicable to this 
section.) Thus, in measuring a fast-neutron flux distribution by ob- 
serving the proton recoil range spectrum, it is necessary to set up strict 
criteria for the acceptance of tracks. Such criteria usually involve the 
choice of a limiting angle, O of Eqs. (109") and (108a). In observing 
tracks which "dip" in the emulsion, it must be kept in mind that emul- 
sions shrink by a factor ~2 in being processed. 

It is relatively easy to make corrections for the background due to 
other (n,p) reactions in the emulsion — say on N 14 — or due to scattered 
neutrons which strike the emulsion from all directions. Since the recoil 



424 



The Neutron 



[Pt. VII 



protons from neutrons in the beam are confined to the forward hemi- 
sphere and the background protons are usually distributed with approxi- 
mate spherical symmetry, measurement of the proton recoil spectrum 
in the backward hemisphere can provide the necessary data for this 
correction. 

For neutrons of energy greater than ~1Q Mev, an appreciable frac- 
tion of the recoil protons leave the emulsion before coming to the end of 
their ranges; the range of a 10-Mev proton is ~600 n of emulsion. Thus, 
as the neutron energy is increased, an appreciable correction must be 
applied for the loss of tracks. This correction can be decreased by using 

0.3 



^.0.2 



Li 




. 


















































\ 


a a 










J> 




^ s 































0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 

•E„(Mev) 

Fig. 49. Li 6 (n,a)H s cross section for normal lithium (7.4%Li 6 , 92.6%Li 7 ) due to 
J. M. Blair et al. (unpublished) (G20, A2). 



thicker emulsions. For very fast and especially ultrafast neutrons, it is 
more accurate to measure the proton energy by the track's grain density, 
or by the small-angle scattering. 

In the intermediate-energy range, the proton recoil method is limited 
by the short range of the recoil protons, which renders inaccurate the 
measurement of range and of angle. Most experimenters regard 1 Mev 
as a practical lower limit of the neutron energies to which the technique 
is applicable. Recently, Nereson and Reines (Nl) have shown that, by 
careful application of the technique, accurate neutron flux measure- 
ments can be made down to E n = 0.5 Mev, and possibly pushed (with 
poor resolution and large corrections) to 0.3 Mev. 

The use of hydrogen-containing cloud chambers for neutron detec- 
tion is subject to the same considerations as have been outlined above. 

(c) Other Reactions — Especially Li 6 (n,a)H 3 . For the detection of fast 
neutrons by a charged particle reaction in nuclear emulsions, it is neces- 
sary that the cross section for the reaction in the emulsion be at least 
comparable with that of neutron-proton scattering, and that the reaction 
products be distinguishable from the recoil protons. In the energy range 
~0.1-10 Mev, the Li 6 (n,a) reaction, Q = 4.79 Mev, has the desired 



Sec. 3C] 



Sources and Detectors 



425 



properties. The (n,a) cross section of normal lithium (7.4% Li 6 ) is 
shown in Fig. 49 for neutron energies up to 0.8 Mev. 

In order to overcome the large proton recoil background, it is neces- 
sary to employ emulsions impregnated with enriched Li 6 . Such emul- 
sions have been used by Keepin and Roberts (K5) (Eastman Kodak 



">B*' ,1 



' > ''V- . I 

I. 7 •■■ 



\ 



**■• "' \ 



•09 % 



\ 



t 



Wmi§m 



$P$$l)!0&i>'- 



||E 



>V: 







Fig. 50. Photomicrograph of a Li 6 (n,a)He 3 disintegration in a Li 6 loaded NTA 
emulsion. The lower track is due to the alpha-particle. The background tracks 
are mainly due to proton recoils. Courtesy J. H. Roberts (private communication). 

NTA, with 0.04 g/cm 3 Li 6 loading), who have successfully demonstrated 
the usefulness of this method of fast-neutron detection and flux measure- 
ment. 

In the measurements of Keepin and Roberts, the range of the alpha- 
particle and of the triton are measured, as well as the angle between 
them. They have found that the best energy resolution, ±0.1 Mev for 
neutrons from to a few Mev, is obtained by selecting only those events 
for which the angle between the particles is between 175° and 180°. It 
is necessary to use special processing techniques to permit discrimination 
between the alpha-particle and triton tracks. A photomicrograph of a 
typical event is shown in Fig. 50. 



426 The Neutron [Pt. VII 

The above-mentioned resolution of ±0.1 Mev has been achieved for 
collimated neutron sources. With isotropic neutron fluxes, for which 
this method is possible where the proton recoil method cannot be used, 
such good energy resolution has not been attained (R12). 

For neutrons of energy greater than ~5 Mev, the very large back- 
ground of protons of comparable range renders the use of the Li 6 (w,o:) 
reaction relatively inaccurate. However, for E n > 10 Mev, the 
B 10 (n,2a)H 3 reaction has been successfully used for neutron detection 
with boron-impregnated emulsions (L5). 

D. Slow-Neutron Spectroscopy 

Probably more work has been done, and more information obtained, 
on the properties of neutrons between and 10 ev energy than on all 
the rest of the neutron energy spectrum. This was certainly true before 
and during World War II, and continues, to a somewhat lesser extent, 
to be true today. This emphasis has not been entirely due to the greater 
ease of producing and detecting slow neutrons. The existence and the 
properties of distinct, narrow, slow-neutron resonances provide a con- 
firmation and a test of the ideas of Bohr and his followers on the proper- 
ties of the compound nucleus, and on the possibility of a statistical 
treatment of nuclei. Furthermore, the wave properties of the neutron 
and the fact that slow-neutron wavelengths are of the order of inter- 
atomic distances in molecules, liquids, and solids hold forth the intriguing 
prospect of using slow neutrons as a tool for the study of the structure of 
matter. While a number of such studies were carried out in pre- World 
War II days, this aspect of slow-neutron physics is only now, owing to 
the availability of the strong sources from piles, bearing fruit; these 
studies will be the subject of Section 5. 

The history of the development of techniques for slow-neutron investi- 
gations was summarized in Section 2A. In this section these techniques 
are discussed in detail. Although many types of experiments can be 
designed to measure the variation of some effect or other with neutron 
energy, the quantity most frequently measured is the cross section. 
The cross section is usually measured by a transmission experiment. 

A "good geometry" transmission experiment measures the total cross 
section, <r = £) cr,-, the sum of the cross sections for all processes which 

i 

remove neutrons from a beam; these include scattering as well as ab- 
sorption processes. The quantity measured in a transmission experi- 
ment is the attenuation 

y = e~ N ° T (112) 



Sec. 3D] Sources and Detectors 427 

the ratio of neutron intensities at the detector with and without an 
absorber (NT = number of absorber atoms per square centimeter of 
absorber *) in the beam. Such a measurement does not require any 
knowledge of the value of the neutron flux. 

In order to obtain maximum accuracy in a minimum of counting 
time, especially if the available beam intensities are not very great, it is 
necessary to make a proper choice of the absorber thickness. The opti- 
mum thickness, which depends on the details of the experimental 
arrangement and on the background, is, over a wide range of experi- 
mental conditions, for NaT « 1-2 (R17, R9). 

Since scattering is always one of the components of the total cross 
section, it is necessary, in a good geometry experiment, to correct the 
measured transmission for those scattering processes in which the de- 
flected neutron strikes the detector. 2 The smallness of the correction is 
the measure of the goodness of the geometry. In order to make this 
correction, it is necessary to know the magnitude and the angular dis- 
tribution of the scattering cross section. 

Under certain circumstances, a "poor geometry" transmission experi- 
ment can yield independent and useful information. Consider a beam 
of neutrons and an absorber, both large in cross-sectional area compared 
to the detector. Let the detector be placed directly behind the absorber. 
The attenuation, I/Iq, due to the absorber is, in this case, e~ N °' T , where 
a' = <r — ^<x sc (provided that the scattering is symmetrical with respect 
to 90°) since half of the scattered neutrons leave the absorber through 
the back face. In all these considerations we assume that the neutrons 
have a negligible probability of making more than one nuclear encounter 
in the absorber; otherwise, the problem approaches one of diffusion, to 
be considered in Section 4. 

A still more extreme case of poor geometry is that of a point source 
completely surrounded by a (spherical) absorber. In this case, if the 
detector is sufficiently far from the source — so that the distribution of 
scattered neutrons also resembles that of a point source — the meas- 
ured cross section is a a = <r — <r sc . Thus, the results of good and poor 
geometry transmission experiments enable the determination of both 
o- sc and <r = <j — a 8c . Experiments in which the geometry is interme- 
diate between good and poor are rather more difficult to interpret (S65). 

1 If the absorber contains more than one type of nucleus, the exponent of Eq. 
(112) is replaced by NaT = T £ AW. 

k 

2 With respect to this correction, inelastic scattering must be regarded as an ab- 
sorption process if the detector is insensitive to the inelastically scattered neutrons, 
and as a scattering process if the detector is sensitive to them. 



428 The Neutron [Pt. VII 

Transmission experiments are, of course, not the only means of meas- 
uring neutron cross sections. All the techniques whereby specific reac- 
tions can be used for measuring neutron fluxes (discussed in Section 3C) 
can be reversed, and cross sections determined if the neutron flux is 
known. Mention must also be made of the techniques developed by 
Dempster and co-workers (L4) and by Inghram, Hess, and Hayden 
(II, H50, H55) for the detection of (strong) neutron absorption and 
measurement of cross sections by mass-spectrographic observations of 
the changes in isotopic abundances resulting from neutron bombard- 
ment. The reader's attention is especially called to a beautiful photo- 
graph published by Dempster [Phys. Rev., 71, 829 (1947)] showing the 
decrease in abundance of the Cd 113 isotope when normal cadmium is 
subjected to intense neutron irradiation. Finally, a transmission meas- 
urement can be changed into a direct measurement of <j sc by moving 
the detector out of the neutron beam; however, such a measurement 
requires either an exact knowledge of the geometry and the neutron 
flux, or calibration with a scatterer of known cross section, say carbon 
(H38, H40). 

1. Slow-Neutron Monochromators. Devices for singling out neutrons 
of a specific (usually variable) energy are known as monochromators or 
velocity selectors. A number of such devices, based on various princi- 
ples, have been designed and constructed. They all employ, as the 
slow-neutron source, neutrons emerging from a mass of slowing-down 
material (paraffin, heavy water, graphite) whose energy distribution is 
approximately Maxwellian, with an epithermal tail (1/E flux distribu- 
tion) extending into the intermediate-energy range. As neutron detec- 
tor, a boron-containing proportional counter is usually employed. For 
maximum sensitivity over the widest possible range, the counter is filled 
with enriched B 10 F 3 . 

Although the type of monochromator employed depends primarily on 
the nature of the primary neutron source, there are two basic factors 
which determine the usefulness of a given device: (1) the energy range 
over which it can be operated, and (2) the resolution at a given energy, 
which is a measure of the energy spread at the energy under considera- 
tion. In order to define the resolution, it is necessary to specify the 
shape of the resolution function. The resolution function can usually 
be approximated by an isosceles triangle. Unless otherwise specified, a 
triangular resolution function will be assumed, the monochromator 
energy being defined as the energy at the apex, and the "resolution" as 
the energy spread at the base. The characteristics of some of the best 
existing monochromators are summarized in Table 29. The figures in 
the third column for the "time of flight" velocity selectors give the ratio 



Sec. 3D] Sources and Detectors 429 

of the time resolution of the instrument (in microseconds) to the flight 
path (in meters) ; this ratio sets the basic limitation both on the usable 
energy range and on the resolution of the velocity selector. The last 
five columns give the energy resolutions which are normally attained in 
operation. 

TABLE 29 

Characteristics of Some Slow-Neutron Monochromators 





Usable Range 
(ev) 


Resolution 
0xsec/m) 


Resolution (ev) 




0.025 


0.1 


1 


10 


100 


1,000 


Argonne thermal chopper 
(B65) 

Argonne fast chopper (S18, 
S19) 

Columbia modulated cyclo- 
tron (R4) 

Harwell electron accelerator 
(M26) 

Crystal spectrometer (S56) 


0.002-0.2 

0.001- 5,000 

0.001-10,000 

0.001-20,000 
0.02 -100 


(50) 
0.6 
1 
0.5 


0.005 

0.001 

0.001 

0.001 
0.0015 


0.05 

0.05 

0.05 

0.05 
0.01 


0.1 

0.07 

0.1 
0.2 


1.7 
1.2 

1.3 

5 


40 
30 
20 


1,000 

1,000 

500 



(a) Mechanical Velocity Selectors or Choppers. The first slow-neutron 
monochromator, built by Dunning, Pegram, Fink, Mitchell, and Segre 
(D20), was based on the principle used by Fizeau in his classic measure- 
ments of the velocity of light. In this device two cadmium disks were 
mounted, one at each end, on a shaft. Both disks had a series of uni- 
formly spaced radial slits; the disks could be displaced by an arbitrary 
angle with respect to each other. With the disks so misaligned, thermal 
neutrons, moving along the direction of the shaft, which passed through 
the first set of slits were absorbed in the second disk, provided that the 
shaft remained stationary. However, with the shaft rotating, those 
neutrons which passed through the slits of the first disk, whose time of 
flight between disks was equal to the time required for the second disk 
to move through the angle of misalignment, could also pass through the 
second slits and be detected. Thus, by varying the velocity of rotation 
or the angular displacement, it was possible to select neutrons of a pre- 
determined velocity. Although the resolution of this device was poor 
(Sl3a), the velocity selector was useful in demonstrating the 1/v nature 
of a number of thermal neutron cross sections. Improvements and 
further measurements on this velocity selector were carried out by Fink 
(F29). 

Taking advantage of the greater thermal neutron fluxes available 
from the first pile, Fermi, Marshall, and Marshall (F22) constructed a 



430 



The Neutron 



[Pt. VII 



thermal neutron velocity selector based on the same principle, but with 
a number of significant modifications. In this device a beam of neutrons, 
from the thermal column, falls on a cylinder whose axis is perpendicular 
to the beam direction. The cylinder is made up of alternate, lengthwise, 
thin layers of aluminum (effectively transparent to thermal neutrons) 
and cadmium, as shown in Fig. 51. Neutrons can pass through the 
cylinder only when the layers are parallel, within ±3°, to the beam 
direction. Thus, when the disk is rotated (at speeds up to 15,000 rpm) 
about its axis, neutrons are transmitted in short bursts. 



Aluminum v\^ 



Steel 



W 



Cadmium 1 



Multiple sandwich 
of 0.004" to 0.008" 
cadmium and 



aluminum foils 




Fig. 51. Cross section, perpendicular to the axis of rotation, of the mechanical 
"chopper" used in the thermal neutron velocity selector of Fermi, Marshall, and 

Marshall (F22, B65). 

The neutrons which get through the shutter are detected in a bank of 
BF3 proportional counters, 1.46 meters away. The neutron time of 
flight between shutter and detector is determined electronically, by 
allowing the neutron detector to be sensitive only for short periods, 
delayed by an arbitrary time interval t with respect to the shutter-open 
times. The synchronization between the shutter position and the de- 
tector-on time is achieved by attaching to the cylinder axis a mirror 
which reflects a beam of light into a photocell when the shutter has a 
given orientation. By varying the position of the photocell, the delay 
time t can be chosen at will. Only those neutrons will be recorded whose 
time of flight between the shutter and the detector is equal to t. If the 
burst time duration and detector-on times are both equal to At, and 
both have square shapes, the velocity resolution function is a triangle 
with apex at the velocity v = L/t and base width Av = 2v At/t = 
2v 2 At/L, where L = the length of the flight path. 



Sec. 3D] 



Sources and Detectors 



431 



This velocity selector was subsequently improved by Brill and Lich- 
tenberger (B65), who used it to measure the cross sections of a number 
of elements in the thermal neutron region. The numbers given in the 
first line of Table 29 apply to the improved model. 

Velocity selectors which employ cadmium in the neutron shutter 
cannot be used for energies above the cadmium cut-off. This limitation 




Pig. 52a. The fast chopper of Selove, at the Argonne Laboratory (S18, S19). Pho- 
tograph shows the stator, rotor, and photocell (box) for timing. 

does not apply to the "fast chopper" constructed by Selove at the Ar- 
gonne laboratory (S18, S19), which employs, as a neutron shutter, a 
long cylinder of steel with thin slits cut into the cylinder parallel to the 
axis, which is also the direction of the neutron beam. In line with this 
cylinder, which can be rotated about its axis, there is a stationary cylin- 
der with identical slits. Neutrons can pass through both sets of slits 
only when the two sets are aligned; when the slit systems are misaligned, 
the thickness of steel is sufficient to remove neutrons of all energies from 
the beam. This device is shown in Figs. 52a and 52b. 

The neutron detection and timing system is, in principle, the same as 
that of the thermal chopper. In operation, the flight path L = 10 
meters and the burst length A* « 6 n sec. The characteristics of this 
velocity selector are given in the second line of Table 29. The possibil- 
ity of using such a relatively high resolution device is dependent upon 



432 



The Neutron 



[Pt. VII 



Slit width (W) 




Fig. 52b. Details of the rotor construction of the chopper in Fig. 52a. 



1.0 



0.5 













"f!'^> 


Y\ 


J ^A { • 




T\ l?~-c 


r ■• "\o/^~7 








W 182 





a i.o 



0.5 




1.0 



0.5 



10 



20 
IX sec/m 



30 



g/cm 2 

w 182 W 183 w 184 w 186 

1.58 




0.054 0.07 0.021 



1 




o 


°„ 


N^ 


>*~~«» 


V-S. 




o ° 






V 


W 186 





007 0.006 0.023 













X ^\ 


AH 


^ * , 


f\! 


•\ L>-< 


\ 


1 ' 


fv/ 


\J 

Normal 
wolfram 





8.04E/cm 2 242 240 

(3Q.H)(29.8») 
(22JJ) j 39 
U7-3J) 



5 

5000 500 

1 1 1 i — i — i — i — i — i 1 — i — i — i — r 

1000 200 100 50 30 20 15 10 8 6 5 4 



Fig. 52c. Transmission curves for separated wolfram (W) isotopes, made with the 

fast chopper of Fig. 52a. 



Sec. 3D] Sources and Detectors 441 

low-energy side of the resonance never falls below 2000 barns. On the 
high-energy side, on the other hand, the cross section descends abruptly; 
above ~4 ev the cross section (5 barns) is due entirely to scattering' 
there being no additional resonances observed up to ~100 ev. A thick- 
ness of 1 g/cm 2 (~l mm) of cadmium metal will transmit less than 1 
percent of all the incident neutrons of energy below 0.3 ev, and ~M of 
the neutrons of energy 0.45 ev. 

Only a negligible portion of the Maxwell distribution (for reasonable 
neutron temperatures) extends into the epicadmium region. On the 
other hand, if the absorber has a resonance fairly close to the thermal 
region, the low-energy tail of the resonance may extend to below the 
cadmium cut-off; in this case, the absorption of a cadmium-covered de- 
tector will be less than the true resonance absorption, and it is neces- 
sary to apply a correction (increase) to the cadmium-covered detector 
reading before subtracting it from the bare detector value, in order to 
obtain the true thermal neutron absorption. The magnitude of this 
correction will depend on the thickness of the cadmium shield. 

Indium is frequently used for a thermal neutron detector because of 
the magnitude of its cross section (<J ^ 190 barns), the ease of handling 
indium metal foils, and the convenient half-life of the resulting radioac- 
tivity (54 min). The lowest indium resonance is at 1.44 ev (<r ^ 35,000 
barns, T s 0.08 ev; see Fig. 7c, page 281). The correction to the activ- 
ity of cadmium-covered indium foils to obtain the true resonance activa- 
tion has been investigated experimentally by Kunstadter (K26), who 
gives the following correction formula: 

True activity = observed activity X e 0A3ST (115) 

where T is the thickness of cadmium metal in millimeters. 1 

(6) The Pile "Danger Coefficient" Method. A useful and accurate 
method of measuring thermal neutron absorption cross sections has been 
reported by Anderson, Fermi, Wattenberg, Weil, and Zinn (A25). This 
method involves the use of a thermal pile (see Section 4), and depends 
on the fact that introduction of absorbing material into the pile causes 
a decrease in the pile reactivity. If the pile is operating at a constant 
neutron flux level, the insertion of the absorber will result in a gradual 
decrease in the flux; the effect of the absorber can be compensated by 
displacing the pile control rods by an amount sufficient to maintain the 
pile at a constant power level. 

If the compensating motion of the control rod is calibrated against a 
series of absorbers of known cross section, the required motion of the 

1 Below ~0.5 mm of cadmium, the shield is no longer black to thermal neutrons, 
and the above considerations break down. 



442 The Neutron [Pt. VII 

control rod for an unknown absorber provides a measure of the un- 
known absorption cross section. This method was first devised in order 
to ascertain the harmful effects (with respect to pile reactivity) of various 
contemplated pile construction materials; hence the term danger co- 
efficient. 

One of the characteristics of this method is that it measures the 
absorption cross section, in contrast to the total (absorption plus scat- 
tering) and the activation (absorption leading to a measurable radio- 
activity) cross sections. The pile reactivity is relatively insensitive to 
the scattering properties of the absorber, even if the absorption cross 
section is comparatively small. Since the products of the particular 
absorbing reaction (provided they are not neutrons) are of no concern 
to the pile, this method provides one of the few means of observing ab- 
sorptions which lead to a stable product nucleus, or to a product nucleus 
whose radioactivity is difficult to observe; for example, although activa- 
tion measurements (S24) seemed, at first, to yield an apparent absorp- 
tion cross section of 0.02 barn for niobium, which has the single isotope 
4iNb 03 , the danger coefficient method gives a value of 1.4 barns (A25). 
The discrepancy was resolved when a previously unrecognized isomer of 
Nb 94 was discovered. 

A modification of the danger coefficient method suggested by Wigner 
greatly increases its sensitivity. This is the method of the "pile oscilla- 
tor," in which the absorber is intermittently introduced into the pile, 
resulting in an oscillation of the flux (power) level of the pile. The 
magnitude of the resulting pile oscillation is a measure of the absorption 
cross section of the sample. The effect of scattering by the sample is to 
introduce an out-of -phase component into the pile oscillation; accord- 
ingly, the effects of scattering and absorption can be separated experi- 
mentally, and the absorption of samples for which the absorption cross 
section is only a small fraction of the scattering cross section (e.g., bis- 
muth) can be measured. 

The theory of the pile oscillator has been developed by Weinberg and 
Schweinler (W12). Extensive pile oscillator measurements of thermal 
neutron absorption cross sections have been made at the Argonne 
Laboratory by Harris, Muehlhause, Rasmussen, Schroeder, and Thomas 
(H42), who used an oscillator constructed by Langsdorf (L3), and at 
the Oak Ridge National Laboratory by Pomerance (P22, P23) with the 
oscillator of Hoover, Jordan, Moak, Pardue, Pomerance, Strong, and 
Wollan (H65). A pile oscillator has also been developed and used with 
the French heavy water reactor by Raievski and Yvon (R2). 

In using a thermal pile for a danger coefficient measurement of a 
thermal neutron absorption cross section, it is necessary to correct for 



Sec. 3D] Sources and Detectors 443 

the absorption of an appreciable epithermal neutron component. This 
correction can be determined by the cadmium difference method. 

The danger coefficient method can also be applied to the measurement 
of epithermal absorption cross sections (cadmium-covered foils) (L3). 
It can be used with intermediate- and fast-neutron piles. In these cases 
the absorption is an average over the relatively broad neutron flux dis- 
tribution in the pile. However, in these applications it is much more 
difficult to correct for the effects of the scattering properties of the 
sample. 

3. Measurement of the Characteristics of Slow-Neutron Resonances. 
One of the most important problems of slow-neutron spectroscopy is the 
precise determination of the constants associated with resonances. In 
the following discussion, we consider only capture resonances, neglecting 
scattering and associated interference effects; i.e., we assume r « r r 
J£> r„. (Similar considerations can be applied, with minor modifications, 
to scattering resonances.) In this case, the Breit-Wigner formula is 
most conveniently written 

(E r /E)% 

° m - m-^-wn <n6) 

A resonance is completely described in terms of three parameters: E r , 
the resonance energy; o- , the peak cross section; T, the full width at 
half maximum. 

Equation (116) is not symmetrical about the energy E r , owing to the 
factor (Er/E)^. The variation of this factor is important for resonances 
occurring in or near the thermal region. However, for resonances for 
which T « E r , the deviation of the factor (E r /E) 1A from 1 is negligible 
over the region of significant values of v(E) ; this situation prevails for 
practically all the observed resonances for which E r ^ 1 ev. 

(a) Monochromator Measurements. In principle, a measurement of 
u{E) vs. E provides all possible resonance data. In practice, the finite 
resolution of the monochromator has a profound influence on the ob- 
served shape of the resonance, and on the possibility of deducing the 
resonance parameters. In the following considerations of the effects 
of finite resolution we assume a rectangular resolution function, of 
width R, mainly for ease of computation. The computations can be 
performed for triangular, or any other shape, resolution functions by 
numerical iteration if necessary; furthermore, a resolution function of 
arbitrary shape can usually be approximated with reasonable accuracy 
by the superposition of a number of rectangles. 

Since monoehromators measure <r(E) by transmission experiments, the 
absorber thickness is important to these considerations. We define (1) 



444 The Neutron [Pt. VII 

a thin absorber as one which is at least translucent over the entire 
resonance, i.e., NT<r < 3; (2) a thick absorber is black in the region of 
the resonance, NTcr > 10. It should be remarked that the choice be- 
tween using a thin or a thick absorber is not entirely a question of the 
available techniques of absorber preparation. The monochromator 
resolution is the determining factor: for R » T it is impossible to obtain 
sufficient absorption, and consequently to make any measurement at 
all, by using a thin absorber. On the other hand, for R < r the use of 
thin absorbers is, as we shall see, possible and desirable. 

The available monochromator resolutions have been summarized in 
Table 29. Since resonance widths are, for heavy nuclei, ~0.1 ev, good 
resolution (R < T) is possible only below ~1 ev; in the region 1-10 ev, 
the available resolutions are fair; above ~10 ev, resolutions are poor. 
The situation with respect to scattering resonances in medium nuclei is 
somewhat better; however, very few such resonances fall in the slow- 
neutron region. The considerations of this section can also be applied 
to measurements in the intermediate and fast regions, where many scat- 
tering resonances have been observed. 

(1) Thin Absorber Measurements. For a sufficiently thin absorber, 
the transmission is determined by the average, over the resolution func- 
tion, of the cross section. Neglecting the variation of (E r /E) V2 , 

.E+RI2 



&(E) = - I o{E) dE 

R Je—R/2 



itan 

2R 



2(E - E r ) R 

r rj 



— tan 1 



2(E - E r ) R\ 

r rl 

(116a) 



Curves of a/cr are plotted in Fig. 56 as a function of 2(2? — E r )/T, for a 
number of values of R/T. For R/T < 1, the main effect of the finite 
resolution is to depress the peak without greatly broadening the reso- 
nance (although the observed width at half-maximum is somewhat in- 
creased). Thus, for good resolution experiments, it is relatively easy to 
make the appropriate corrections to the transmission curves, and to 
determine the resonance parameters. (The work of McDaniel, Mil, 
on the 1.44 ev resonance of indium, see Fig. 7d, illustrates the method.) 
However, for R/T > 1 the observed width is essentially the resolu- 
tion. It is impossible, from such a measurement, to determine sepa- 
rately the resonance width or the peak cross section; a poor resolution 
thin absorber experiment can only yield the product a r, through 



Sec. 3D] 



Sources and Detectors 



445 



















1 


G 
























































































w* 












































i-4 










































/ 


»-H 




















/ 




















7 


5 

4 


/ 


i— 1 


















^ / 






















/ 












































II 
























*!•-. 




















CN 












^P 


o 

— II — 


10 
© 


»H 






«|t, 














II 


II 


















CM 



























i? 


fc) 


CD 






1 






^^ 


K) 


cr 


-*3 


W 


cj 



\ 



K) 



-"•o/a 






5. 2 


o 




« 




r 1 


!> 


bXJ 


r> 


cm o 






Hi 








cl 


CJ 


a; 


IT 




e 




•n 
















cd 


!> 


^3 


03 


(H 


o 

CO 


Ph 


CD 
> 


,0 




O 


01 


"5 




a 


br 




:a 





U 


+3 




T 




cu 




(3 


o 


1*1 




a 




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446 The Neutron [Pt. VII 

either the observed peak cross section or the area under the observed 
resonance: 

(a) The apparent (observed) peak cross section J is 

r _.B 

c{E r ) — <r — tan — 
R T 



R 
for - « 1 

r 



-HOT] 

r /ir r\ r 

"""r^-r) ""t >>1 (1I6a,) 

The reduction of the peak cross section (by the factor 1.571 T/R) makes 
it virtually impossible to use a thin absorber when R/T > 10. 

(b) It may be seen, by visual inspection of Fig. 56, that the decrease 
in the peak cross section is accompanied by a raising of the tail of the 
observed resonance, so that the area underneath the resonance curve 
appears relatively insensitive to the resolution. Actually (from Eqs. 
116 and 116a) 2 

fff(E) dE = fff(E) dE = - (T r (116b) 

independent of R ! 

Another possible effect of poor resolution is the failure to resolve close 
resonances. This difficulty is illustrated in Fig. 57, in which o/oq has 
been plotted, for two identical resonances separated by the energy D, 
for a number of values of R/T. The classic example of the phenomenon 
of close resonances is the case of iodine, which appeared, from early 
experiments, to have a single, anomalously broad resonance at ~40 ev. 
Careful investigation by Jones (J 12) established the existence of a num- 
ber of sharp resonances between 20 and 50 ev. Many of the relatively 
broad peaks in cross-section curves (G20, A2) observed above ~10 ev 

1 Another resolution shape, for which the average can be performed analytically, 
is the Gaussian function of half-width R. In this case, 

HEr) = <r [1.665(r/fi)e o.693rV-R 2 erfc (0.833l7#)] 

S 1.476<r r/ft for R/T » 1 (116a") 

in which erfc (x) = (ir/4)* -JJe-^dt. 

2 The integrals are conventionally and most easily performed between the limits 
— x and M. Since the entire contribution is for values relatively close to E r , this 
extension of the limits of integration introduces a negligible error. 



Sec. 3D] Sources and Detectors 447 

in monochromator measurements on heavy nuclei are probably due to 
the failure to resolve such groups of resonances. 

(2) Thick Absorber Measurements. If a thick absorber is employed 
in a good resolution experiment, the transmission is ~0 over the main 
portion of the resonance; only the wings of the resonance can be studied 
in any detail. However, as we have seen above, thick absorbers are 
necessary in a poor resolution experiment to obtain transmissions which 
differ significantly from 1. 

The interpretation of the transmission curve for a thick absorber in a 
poor resolution monochromator experiment can be achieved by numeri- 
cal and graphical techniques, as. described by Havens and Rainwater 
(H45). The result of such interpretation — either by detailed fitting of 
the transmission curve or by integration of the absorption over energy 
— yields a value of the combination a T 2 . The interpretation requires 
a knowledge of the value of the resolution width, R, and of the shape of 
the resolution function; the results are, however, not very sensitive to 
the exact form of the resolution function. 

The fact that the combination a^T 2 determines the results of a poor 
resolution thick absorber transmission measurement can be seen by the 
following argument: The mean transmission at the energy E is 

I/I (E) = ( C R {E' ! E)e' NTn/n+i( - E '- IS ' )Vri] dE'\R(E) (117) 

where R(E',E) is the resolution function; the resolution width is de- 
fined as 

R(E) = fR(E',E) dE' (117') 

Assuming NT<r > 10, the integrand vanishes unless 4(£" — E T ) 2 /Y 2 
^> 1 ; hence, the integral may be replaced by 

I/h(E) ^ ( CR(E',E)e- NT " oTi/ ^ E '- Er) dE'\R(E) 

= F{E,E r ,a Y 2 ) (117a) 

The area under the absorption curve is 



/' 



(1 - ///„) dE = F'(E r ,a T 2 ) (117b) 

Since almost all monochromator measurements for E ^ 50 ev employ 
poor resolution and thick absorbers, a resonance in this region must be 
comparatively strong in order to be observed at all. There is no doubt 
that such measurements fail to detect a fair fraction of the resonances. 



448 



The Neutron 



[Pt. VII 





















i 


i 


















T-t 

II 






















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J 




















ii 


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10 






















II 






























































\ 



ts 



I 



O OS 

r-i O 



to 
o 



o o 



Sec. 3D] 



Sources and Detectors 



449 



















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X > 
















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11 


o=^ 














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O © © 



8=1 E 



E 



450 The Neutron [Pt. VII 

Many of the observed peaks (G20, A2) are probably due to resonance 
groups, as in the case of iodine. 

Finally, it should be noted that considerable progress has been made 
in the direct application of slow-neutron monochromators to the study 
of the details of resonance scattering, by Tittman, Sheer, Rainwater, 
and Havens (T15). The interpretation of measurements on the scat- 
tered neutrons is quite complicated, but it follows, roughly, the tech- 
niques outlined above. 

(b) Self- and Mutual-Indication Techniques. In the pre-velocity 
selector era of slow-neutron physics, a number of techniques were de- 
veloped for studying resonances from which it was possible to deduce the 
resonance parameters, with surprising accuracy in some cases. While 
some of these techniques have been rendered obsolete by more recent 
developments, others still find useful application in resonance investiga- 
tions. Unfortunately, in the first rush to exploit the new-found wealth 
of velocity selectors and pile intensities, some of these old methods have 
been disparaged, neglected, or forgotten. Their revival should con- 
tribute useful information to the growing fund of resonance data. These 
techniques are admirably summarized in the famous "Bethe Bible" 
(B24). 

One of these is the method of self-indication, in which the resonance 
under investigation is used as its own detector (F16). For a thin reso- 
nance detector in a slow-neutron beam, the saturated activity is (from 
Eqs. 103'b, 104, 104a, 103c) 



»© 



R = NTA<t>(E r ) ( - 1 «r r (103d) 

Thus, knowledge of <r r will directly yield the flux <f>{E r ), and vice versa. 1 
If now a thin absorber of the same material (thickness T') is placed 
in the beam, the saturated activity of the detector will be reduced, 
becoming 

R' = NTA f[<j>(E) - NT'^(E)a(E)]o-(E) dE 



R [l - NT' fa 2 dE/ fa 



dE] 



= R [L - NT'ao/21 (118) 

'We assume, here and in the following, that a = a{n,y); i.e., r ~ l\^i> l'„. 
This holds, of course, only for capture resonances in heavy nuclei. The necessary 
modifications, if this is not the case, can be obtained by arguments similar to those 
employed in this section. 



Sec. 3D] Sources and Detectors 451 

Thus, the effective cross section for a thin resonance absorber in a self- 
indication experiment is a /2. 

Thus, it is, in principle, possible to measure <r without a monochroma- 
tor. There are, however, a number of difficulties inherent in the use of 
this technique. In the first place, since cr can be very large (~10 4 -10 5 
barns for resonances at a few ev energy in heavy nuclei), the prepara- 
tion of thin absorbers may present serious technical problems. The 
common solution is to use absorbers of decreasing thickness, and to 
extrapolate the resulting values of <r to zero absorber thickness. 

Another difficulty arises from the possible activation of the detector 
by neutrons of energies outside the resonance. Thermal neutrons are 
easily excluded by cadmium. Other resonances are more difficult to 
eliminate, especially if they should happen to be of comparable strength 
(<r r) to the one in question. Fortunately, this is seldom the case; the 
lowest resonance usually dominates the rest so that, especially if ab- 
sorber and detector are truly thin, the observed effects can be attributed 
to a single resonance. It should be noted that the availability of a mono- 
chromator of even poor resolution would eliminate the effects of other 
resonances in a self-indication experiment. However, only the crystal 
spectrometer can be used in this connection, since other velocity selec- 
tors do not remove neutrons from the beam. 

When the resonance is used as its own detector, the resonance energy 
can be determined by measuring the cross section of boron for the de- 
tected neutrons. Since the cross section of boron follows the 1/v law, 
and its absolute value is known, this method can yield an accurate 
value of E r . However, the 1/v dependence of the total (transmission) 
boron cross section holds only as long as the absorption cross section is 
much greater than the (constant) scattering cross section. This limits 
the, use of normal boron to energies below ~100 ev; for absorbers en- 
riched in B 10 , the range is considerably extended. 

An interesting example of the use of these techniques for the study of 
a single resonance (rhodium, E r = 1.2 ev) is the work of Manley, Gold- 
smith, and Sch winger (M3). These authors also used the shape of the 
curve of transmission vs. (self-indicated) absorber thickness to obtain a 
measure of the resonance width, r. However, as the absorber thickness 
is increased, the effects of other resonances are of greater importance. 

A natural extension of the technique of self-indication can be made 
for the case of two resonances (in different materials) which partially 
overlap; one can be used as absorber and the second as detector, thereby 
obtaining a measure of the area of overlap of the levels; this is the method 
of mutual indication. By measuring all four absorption coefficients (two 



452 The Neutron [Pt. VII 

self, two mutual) it is possible to obtain the relative widths and reso- 
nance energies for the two levels (B24, A20, F16, H66). 

The techniques described above, with a number of ingenious modifica- 
tions, have been extensively investigated and exploited by a group at the 
University of Groningen, Holland (C27, C28). The technique of 
mutual indication has recently been applied by Hibdon and Muehlhause 
(H57) to the detection of resonances in a number of substances; they 
used the scattering resonances of cobalt and manganese as indicators. 

(c) Interpretation of Resonance Data; the Doppler Effect. The fore- 
going discussion may be summarized as follows: There are available a 
number of techniques for the measurement of the constants which char- 
acterize a single resonance (Eq. 116). Monochromator measurements 
determine E r and, for good resolution and thin absorbers, <r and I\ 
Poor resolution monochromator measurements yield o- r for thin ab- 
sorbers, or cor 2 for thick absorbers. Self-indication measurements 
yield o- and <r r (if the neutron flux can be measured independently), as 
well as E r and, possibly, I\ Mutual-indication measurements, in con- 
junction with another resonance of known characteristics, can give E r 
and T. 

There is still another method for obtaining information concerning a 
resonance, provided that it is permissible to extend the single resonance 
formula into the thermal region; this is possible if the resonance in ques- 
tion is the strongest of all the resonances of the nucleus in question, and 
also has the lowest energy. For the thermal energy region, if E th = E 
<3C E r , the one-resonance (n,y) cross section is 

"-^^W (119) 

which is the usual 1/v law. Thus, the value of the thermal neutron 
capture cross section determines the constant <r T 2 , if E r is known. We 
recall that, for a 1/v absorber in a Maxwellian neutron distribution of 
temperature T, 

(T t h = (95a) 

1.128 

whence 

(t T 2 ^ 1.128cf„,- ±E/ A (kTy A (119a) 

If this method is used to derive <x T 2 from the thermal neutron capture 
cross section of In 115 (Table 10, page 328) the value obtained, ~185 
barns-ev 2 , agrees, to within -~10 percent, with that computed from the 
constants of the 1.44-ev resonance as measured by McDaniel (Mil). 



Sec. 3D] Sources and Detectors 453 

The connection between am and the first resonance depends on the 
possibility of neglecting the effect of all higher resonance. The resonance 
energy factor in Eq. (119) favors this possibility. The value of or , 

.0 = ^^ (120) 

is also a decreasing function, ccE r ~ l/i , of the resonance energy, provided 
that the resonances are all of the same type; i.e., T nri ^ CE ri ^ and I\ 
= constant. 

Equation (120) also shows that the neutron width, fT nr , can be ob- 
tained from a measurement of <r and T (actually, from col 1 ). 

In the preceding discussion, r has been associated with the true 
width of the resonance. Actually these are not identical, owing to the 
Doppler effect of the thermal motion of the absorber nuclei. The Dop- 
pler effect is equivalent to introducing an inherent finite resolution into 
any measurement, since an incident neutron of energy E will have a 
spread of energies relative to the absorbing nuclei. The "resolution 
function" which describes the Doppler effect is 

R D {E',E) = e -w-*)VA» (121) 

where „ 

A = 2 I J (121a) 

and A is the mass number of the absorbing nucleus. The Doppler 
width, A, is by no means negligible. For E = 1 ev, kT — 34o ev > A. 
= 100, Eq. (121a) gives A = 0.032 ev; A increases like E y \ The effec- 
tive cross section of a given absorber is 



with 



, ff (E) = (fa(E')R D (E',E) dE'\R D (E) (121b) 

R D (E) = (r d (E',E) dE' = t a A(E) (121c) 



Any additional effects due to the finite resolution of the measuring de- 
vice are superimposed on the Doppler broadening. 

The result of this Gaussian resolution function is to reduce the peak 
cross section and broaden the resonance in the manner previously de- 
scribed. 1 The details of the Doppler broadening were derived and dis- 
cussed by Bethe and Placzek (B25, B24), who give formulas and curves 
for the interpretation of the measured constants (also in self-indication 
experiments) in terms of the true constants of the resonance. 

1 See footnote 1 on page 446. 



454 The Neutron [Pt. VII 

(d) Effects of Many Resonances; the Resonance Integral. Some medium 
and most heavy nuclei have many resonances in the slow-neutron region. 
The absorption, scattering, and activation of these nuclei in a slow- 
neutron flux will be due to the combined effect of all the resonances. 

Consider a thin detector of a given type of nuclear reaction, with cross 
section <ri(E). In the unusual 1/E slow-neutron flux, with the thermal 
neutrons eliminated by cadmium, the total number of processes of type i 
will be (Eqs. 96, 103c", 104b) 

Ri = Nvf 4>(E)<Ti(E) dE = NVB f a(E) ( — ) (122) 

J~0.5ev J~0.5ev \E/ 

If <ri(E) is characterized by a number of resonances, their effect is sum- 
marized by the resonance integral 



Recalling that, for a process of type i, 






we have 

S^2x 2 E- 2/j ' r " ryIV (123a) 

j TjE r j 

Resonance integrals can be evaluated, given a 1/E epicadmium flux, 
in a straightforward fashion. The activation of an (n,y) detector gives 
2„ for the resonances leading to the observed activity. Pile danger 
coefficient measurements can yield S B for all capture processes, includ- 
ing those which lead to undetectable product nuclei. Measurements of 
the total scattering cross section yield S s , provided that a reliable means 
can be found for subtracting the effect of the relatively constant poten- 
tial scattering. 

Harris, Muehlhause, and Thomas (H40) have reported measurements 
of S a and S s for a number of nuclei. They measured S a by observing 
the activation of thin detectors in the 1/E flux next to the Argonne 
heavy water reactor. Absolute values were obtained by comparison 
with the thermal neutron activation of the same detector, using the 
known thermal neutron cross sections and correcting for the effect of 
1/v absorption above the cadmium cut-off. The ratio of thermal to 
epicadmium flux was determined from the known values of a ath = 93 



Sec. 3E] Sources and Detectors 455 

barns and S a = 1296 barns (due almost entirely to the resonance at 
~5 ev) of gold. 

Values of S s were measured by the scattering of thin samples placed 
at the center of an annular BF 3 counter. Absolute values were obtained 
by comparison with the scattering of a carbon target, for which the 
average epicadmium cross section (4.60 barns) was independently deter- 
mined. Corrections were computed for the (constant) potential scatter- 
ing, when known. In other cases, the potential scattering contribution 
was measured by using a neutron beam which had been filtered through 
a thick absorber of the same material as the scatterer (a sort of self- 
indication in reverse). Unfortunately, the scattering chamber efficiency 
depended on the energy of the scattered neutrons, so that the interpreta- 
tion of the results required a knowledge (or assumption) of the energies 
of the most important — i.e., first few — scattering resonances. 

The results of these measurements were interpreted in terms of aver- 
age values of V n /T for the nuclei in question (see Section 2). From the 
energy dependence of the factors in Eq. (123a) it is seen that the par- 
ticular average, deduced from 2 3 /(S a + S s ), is strongly weighted in 
favor of the lowest-lying resonances. 

E. The Calibration of Neutron Sources 

Along with the large variety of available neutron sources with dif- 
ferent spectra and yields, there have been devised a number of methods 
for measuring source spectra, some of which have been described in the 
preceding, and for measuring source yields — the subject of the following 
discussion. Some of these methods are quite specific to the reaction 
under investigation. Thus, for charged particle reactions on light nu- 
clei, the recoil (product) nucleus can frequently be detected and counted 
by conventional means, thereby providing a measure of the neutron 
yield. An example is the measurement of the yield of the reaction 
H 3 (d,n)He 4 by counting the recoil alpha-particles (T2). 

Alternatively, if the product nucleus is radioactive, the yield of a 
reaction can be obtained from the resulting radioactivity of the target. 
This technique requires, however, a determination of the absolute value 
of a radioactive decay rate; such measurements are, in general, difficult 
to perform with good accuracy. This method is applicable to some of 
the threshold (p,n) reactions. 

An important variation on this theme is the method of Paneth and 
GKickauf (P2, G8) for obtaining the yield of the Be 9 (7,n)Be 8 -> 2« 
source by measuring the total helium accumulation in the beryllium. 
This method could serve as the basis for the calibration of the standard 
neutron source (C42). 



456 The Neutron [Pt. VII 

Another general technique for the measurement of source strengths is 
the observation of the excitation of a reaction of known cross section by 
the source neutrons. This method requires the use of a detector of 
known efficiency and, clearly, is applicable only when the neutrons are 
either monoenergetic or have a relatively simple, known spectrum. The 
most useful reactions for this application are neutron-proton scattering, 
for fast neutrons, and the B 10 (n,a) reaction, for intermediate and slow 
neutrons. 

Most of the useful laboratory sources — radioactive (a,n), photoneu- 
tron, and many accelerated charged particle reactions — have complex 
spectra and are, in general, not susceptible to the above-mentioned tech- 
niques. The calibration of such sources is usually achieved by the 
method of "space integration." There are almost as many variations 
on this technique as there have been applications thereof. However, 
the salient features can be described as follows : 

Consider a source of fast or intermediate neutrons in an infinite slow- 
ing-down medium. 1 Let the source strength be Q (neutrons/sec), the 
density of the medium be N (nuclei/cm 3 ), and the absorption cross sec- 
tion of the medium be a- a (E). At equilibrium, we have 

Q = 4x I r 2 dr\ N<j>{r,E)<r a {E) dE 



ir J r 2 dr I 






= 4tN I *(r)o- a (r)r 2 dr (124) 



where r is the distance from the source. The problem of absolute source 
calibration reduces to the experimental determination of the function 

*(r)fa(r). 

Now, the slowing-down media most commonly used (e.g., water, 
paraffin, graphite) have the property that <r a (E) obeys the 1/v law; 
there are no known absorption resonances up to at least a few Mev. 
Furthermore, the thermal neutron absorption cross sections are small, so 
that the absorption during slowing down is negligible; practically all the 
absorption takes place after the neutrons have reached thermal equi- 
librium. Thus, it is possible to measure $(r)o- a (r) by using any thermal 
neutron detector whose absorption also follows the 1/v law and for 
which the ratio cr a (medium)/o- (detector) is known. (Since both medium 

1 The practical definition of infinite is: large enough so that not more than a small 
fraction of the neutrons escape through the outer boundary; actually, if the geom- 
etry is sufficiently well defined so that it is possible to compute the probability of 
escape, this requirement can be relaxed. 



Sec. 3E] Sources and Detectors 457 

and detector absorb according to the 1/v law, the ratio of their absorp- 
tion cross sections is independent of the position in the medium.) 

As an example consider a small, thin BF 3 counter of known efficiency 
(i.e., known N'V). The counting rate of this detector at the position r 

in the medium is 

R'(r) = N'V'*(rW(r) (103b') 

Combining (124) and (103b'), we have 

*-(£)£)£*<*•* 

It is frequently more convenient to use a radioactive (n,y) detector 
for the flux measurement. (Manganese and indium foils, using the 
cadmium-difference technique, are convenient 1/v thermal neutron de- 
tectors.) In this case, a calibration of the detector efficiency is necessary 
in order to convert measurements of saturated activity to neutron ab- 
sorption rates. Such calibrations can be made if the efficiency of the 
radioactivity counter is known for the particular foils used. Greater 
accuracy of calibration can usually be achieved by comparing the 
saturated activity of the foils used with the counting rate of a thin BF 3 
counter, of known efficiency, in the same thermal neutron flux; this 
method was developed by Frisch, Halban, and Koch (F48). 

An interesting technique of foil calibration 1 was carried out by Seidl 
and Harris (S16). In this method the absorption rate of a thin boron 
absorber, exposed in a thermal neutron flux, is determined by measuring 
the accumulation of He due to theB 10 (n,a) reaction. The saturated 
activity of the foil used in the integration is measured in the same flux. 
This method does not depend on knowing the efficiency of a BF 3 coun- 
ter. It does, however, require the availability of very strong thermal 
neutron fluxes, i.e., a pile. 

A useful and ingenious modification of the above techniques has been 
described by O'Neal and Scharff-Goldhaber (04). Their calibration 
method has the advantages that it requires no knowledge of the values 
of absorption cross sections and the space integration is performed 
physically. The measurement is made in three stages: 

(1) The source is placed at the center of a large tank of water in which 
is dissolved a quantity of a 1/v detector (they used manganese sulfate) . 
After saturation is achieved, the source is removed, the solution 
thoroughly stirred, and the activity measured by immersing a Geiger 

1 Proposed by L. W. Alvarez. 



458 The Neutron [Pt. VII 

counter in the solution, at the center. Let the counting rate be i2 x : 

Ri = af $i(rKWr 2 dr (125a) 

[This method of "physical integration" is described by Anderson, Fermi, 
and Szilard (A23.)] 

(2) Now mix uniformly into the same solution a known amount 
(N'V) of 1/v absorber (they used finely divided manganese powder) 
which can be removed from the solution after irradiation. Irradiate to 
saturation, remove the absorber, and repeat the activity measurement 
on the solution: 

R 2 = a] * 2 W^7(r)r 2 dr (125b) 



Let R = R 2 /Ri- 

(3) It is finally necessary to obtain an absolute measurement of the 
total saturated activity (rate of neutron absorption) of the absorber: 



I = 4irN' I *a(r)«r '(r)r 2 dr (125c) 

This step requires the absolute calibration of the radioactivity-measur- 
ing device and of the absorber sample used. 
Now, since 

Q = MiV Hs0 I ®i(r)<TH 2 o(r)r 2 dr 

r°" 

+ Nunsoi I *iWo-MnS0 4 Wr 2 (ir} 

•'O 

= I + 4n-{2V H2 o I *20>h 2 o(?V dr 
Jq 

+ -^MnS0 4 | *2(»MnS04(» 2 cM (125d) 

and assuming that all absorption is 1/v, which means 

$ 2 W^W = (^) *2«?7« (125e) 

in all possible combinations, it follows that 

Q = ^ (125) 



Sec. 3E] Sources and Detectors 459 

A minor modification and inversion of the same technique, first used 
by Segre (S14), has been used for the most accurate comparisons of the 
absorption cross sections of hydrogen and boron by Whitehouse and 
Graham (W19) and others. Consider first a source in a tank of pure 
water: 

Q = 47riVH 2 o j $o<m^o> 2 dr (126a) 

Now dissolve in the tank some boron of density Nb ; then 

Q = 47t{^h 2 o' J$i^>r 2 dr + N B f^w 2 dr} (126b) 

The activation integral is measured, in each case, with the same 1/v 
detector: 

I = « I $o<Tar 2 dr 

I t = a J Qiw 2 dr (126c) 

Since all the absorbers follow the 1/v law, we can apply Eq. (125e), 
whence 

©■(^) + ©(^) <»> 

The difficult step in any of the many methods of source calibration is 
the absolute determination, at one point, of the rate of absorption of a 
given absorber. Indeed, the various modifications simply shift this 
determination from one stage to another. This difficulty need only be 
solved once, however, for, after the yield of one source is known, the 
yield of any other source can be compared with the standard. 

The methods of comparison are exactly those described above, with 
the exception that the detector does not require calibration. Thus, the 
ratio of the space absorption integrals (Eq. 125c) for two sources in the 
same medium, using the same detectors, is the ratio of the source yields. 
Actually, if the two sources are of the same type — i.e., same dimensions 
and the same energy spectrum — it is merely necessary to compare the 
detector activations at one point in the slowing-down medium. 

In practice, the use of a resonance detector (e.g., cadmium-shielded 
indium) for obtaining the space activation integral is frequently more 
convenient, since the curves of resonance activation vs. r fall off more 
rapidly than for thermal neutrons. Since the activation of a resonance 
detector is proportional to the rate at which neutrons are passing 



460 The Neutron [Pt. VII 

through the resonance energy (see Section 4) and since, for the usual 
moderating media, there is no resonance absorption, the space integral 
for a resonance detector is also proportional to the source strength. This 
technique derives directly from the early experiments of Amaldi and 
Fermi (A20). 

An entirely different method of source comparison and calibration 
can be used if a pile is available. Consider a pile which is exactly criti- 
cal, i.e., there is one neutron produced for every neutron lost. If a 
source is placed in this pile, the neutron density will increase linearly 
with time, since there is no net loss of neutrons in an exactly critical pile. 
The rate of rise of the total number of neutrons in the pile, N, is 

dN 

Tt - Q (127) 

In practice, it is necessary to calibrate the pile and neutron detector, and 
to correct for the absorption by the source. This method is discussed by 
Bretscher (B63). 

Alternatively, if the pile is slightly subcritical, and there is a source 
in the pile, the equilibrium neutron density will be determined by the 
condition that the rate of neutron loss equals the source strength. Since 
the rate of neutron loss from a subcritical pile is proportional to the neu- 
tron density in the pile, the equilibrium density will be proportional to 
the source strength, and it can be used (with proper calibration) as a 
measure of Q. The theory of this method has been discussed by Placzek 
and Volkoff (P18) and by Friedman (F44). 



SECTION 4. THE INTERACTION OF NEUTRONS WITH 
MATTER IN BULK 

A. Introduction 

The preceding sections were devoted to the discussion of the interac- 
tions of neutrons with nuclei and of the instruments and techniques for 
the study of neutron phenomena. In this section we consider the suc- 
cessive interaction of neutrons with nuclei in large masses of material. 
For our present purposes the neutrons can be considered to behave in- 
coherently, their interaction with the medium depending only on the 
densities and cross sections of the constituent nuclei. Coherent inter- 
action phenomena, which depend on the structure of the medium, will 
be discussed in Section 5. 

The subjects of this chapter are: (1) the diffusion of monoenergetic 
neutrons; (2) the slowing down of neutrons; and (3) slowing down and 



Sec. 4A] The Interaction of Neutrons with Matter in Bulk 461 

diffusion in multiplicative media. The treatment is "elementary"; by 
this we mean that simplifying assumptions are made in order to permit 
the handling of problems by well-known mathematical techniques. The 
emphasis throughout is on methods of computation which can be used, 
with comparative ease, in planning and interpreting experiments. These 
methods, although not exact, are sufficiently transparent so as to be 
useful for obtaining physical understanding of and insight into the 
phenomena under investigation. 

The basis principles underlying the mathematic treatments in this 
section were laid down by Fermi (F17), soon after the discovery of the 
"Fermi effect" — the increase of neutron-induced radioactivity resulting 
from the interposing of hydrogenous material between source and detec- 
tor (F14). In general, the diffusion of neutrons in a medium is govern